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

Percentage Accurate: 90.9% → 96.8%

Time: 11.0s

Alternatives: 17

Speedup: 0.7×

• 29.2% 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: 90.9% 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: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \frac{0.5}{a} \cdot t\\ t_2 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t\_1, \frac{y}{\frac{a}{0.5 \cdot x}}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, t\_1, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (* (/ 0.5 a) t)) (t_2 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_2 -2e+265)
     (fma (* -9.0 z) t_1 (/ y (/ a (* 0.5 x))))
     (if (<= t_2 1e+265)
       (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a))
       (fma (* -9.0 z) t_1 (* (* (/ 0.5 a) x) y))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (0.5 / a) * t;
	double t_2 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_2 <= -2e+265) {
		tmp = fma((-9.0 * z), t_1, (y / (a / (0.5 * x))));
	} else if (t_2 <= 1e+265) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = fma((-9.0 * z), t_1, (((0.5 / a) * x) * y));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(0.5 / a) * t)
	t_2 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_2 <= -2e+265)
		tmp = fma(Float64(-9.0 * z), t_1, Float64(y / Float64(a / Float64(0.5 * x))));
	elseif (t_2 <= 1e+265)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(-9.0 * z), t_1, Float64(Float64(Float64(0.5 / a) * x) * y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(0.5 / a), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+265], N[(N[(-9.0 * z), $MachinePrecision] * t$95$1 + N[(y / N[(a / N[(0.5 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+265], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * z), $MachinePrecision] * t$95$1 + N[(N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -2.00000000000000013e265

   1. Initial program 80.1%

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

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/r* N/A

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

      20. lower-/.f64 N/A

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

      21. metadata-eval N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -2.00000000000000013e265 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.00000000000000007e265

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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 97.9

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

      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 97.9

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

   4. Applied rewrites 97.9%

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

   if 1.00000000000000007e265 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 66.0%

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

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/r* N/A

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

      20. lower-/.f64 N/A

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

      21. metadata-eval N/A

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

   4. Applied rewrites 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, \frac{0.5}{a} \cdot t, \frac{y}{\frac{a}{0.5 \cdot x}}\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, \frac{0.5}{a} \cdot t, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+265}:\\ \;\;\;\;\frac{t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, \frac{0.5}{a} \cdot t, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 (- INFINITY))
     (fma (/ y a) (* 0.5 x) (* -4.5 (* (/ t a) z)))
     (if (<= t_1 1e+265)
       (/ t_1 (* 2.0 a))
       (fma (* -9.0 z) (* (/ 0.5 a) t) (* (* (/ 0.5 a) x) y))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y / a), (0.5 * x), (-4.5 * ((t / a) * z)));
	} else if (t_1 <= 1e+265) {
		tmp = t_1 / (2.0 * a);
	} else {
		tmp = fma((-9.0 * z), ((0.5 / a) * t), (((0.5 / a) * x) * y));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y / a), Float64(0.5 * x), Float64(-4.5 * Float64(Float64(t / a) * z)));
	elseif (t_1 <= 1e+265)
		tmp = Float64(t_1 / Float64(2.0 * a));
	else
		tmp = fma(Float64(-9.0 * z), Float64(Float64(0.5 / a) * t), Float64(Float64(Float64(0.5 / a) * x) * y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision] + N[(-4.5 * N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+265], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-9.0 * z), $MachinePrecision] * N[(N[(0.5 / a), $MachinePrecision] * t), $MachinePrecision] + N[(N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 73.8

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

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

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 73.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.00000000000000007e265

   1. Initial program 97.9%

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

   if 1.00000000000000007e265 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 66.0%

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

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

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

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/r* N/A

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

      20. lower-/.f64 N/A

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

      21. metadata-eval N/A

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

   4. Applied rewrites 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 10^{+265}:\\ \;\;\;\;\frac{y \cdot x - t \cdot \left(9 \cdot z\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-9 \cdot z, \frac{0.5}{a} \cdot t, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+265}:\\ \;\;\;\;\frac{t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, \left(-4.5\right) \cdot z, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 (- INFINITY))
     (fma (/ y a) (* 0.5 x) (* -4.5 (* (/ t a) z)))
     (if (<= t_1 1e+265)
       (/ t_1 (* 2.0 a))
       (fma (/ t a) (* (- 4.5) z) (* (* (/ 0.5 a) x) y))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y / a), (0.5 * x), (-4.5 * ((t / a) * z)));
	} else if (t_1 <= 1e+265) {
		tmp = t_1 / (2.0 * a);
	} else {
		tmp = fma((t / a), (-4.5 * z), (((0.5 / a) * x) * y));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y / a), Float64(0.5 * x), Float64(-4.5 * Float64(Float64(t / a) * z)));
	elseif (t_1 <= 1e+265)
		tmp = Float64(t_1 / Float64(2.0 * a));
	else
		tmp = fma(Float64(t / a), Float64(Float64(-4.5) * z), Float64(Float64(Float64(0.5 / a) * x) * y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision] + N[(-4.5 * N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+265], N[(t$95$1 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * N[((-4.5) * z), $MachinePrecision] + N[(N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 73.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 73.8

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

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

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 73.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 1.00000000000000007e265

   1. Initial program 97.9%

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

   if 1.00000000000000007e265 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 66.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 10^{+265}:\\ \;\;\;\;\frac{y \cdot x - t \cdot \left(9 \cdot z\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, \left(-4.5\right) \cdot z, \left(\frac{0.5}{a} \cdot x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ t_2 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+291}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (* 0.5 x) (* -4.5 (* (/ t a) z))))
        (t_2 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_2 -1e+291)
     t_1
     (if (<= t_2 1e+237) (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a)) t_1))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), (0.5 * x), (-4.5 * ((t / a) * z)));
	double t_2 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_2 <= -1e+291) {
		tmp = t_1;
	} else if (t_2 <= 1e+237) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(0.5 * x), Float64(-4.5 * Float64(Float64(t / a) * z)))
	t_2 = Float64(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_2 <= -1e+291)
		tmp = t_1;
	elseif (t_2 <= 1e+237)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(0.5 * x), $MachinePrecision] + N[(-4.5 * N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] - N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+291], t$95$1, If[LessEqual[t$95$2, 1e+237], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -9.9999999999999996e290 or 9.9999999999999994e236 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. clear-num N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 72.4

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

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

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval 72.5

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 72.5

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

   4. Applied rewrites 72.5%

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

   6. Applied rewrites 97.0%

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

   if -9.9999999999999996e290 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 9.9999999999999994e236

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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 97.9

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

      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 97.9

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

   4. Applied rewrites 97.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 10^{+237}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, 0.5 \cdot x, -4.5 \cdot \left(\frac{t}{a} \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot \frac{0.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -2e-118)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e-14)
       (/ (* y x) (* 2.0 a))
       (* (* (* t z) -9.0) (/ 0.5 a))))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) / (2.0 * a);
	} else {
		tmp = ((t * z) * -9.0) * (0.5 / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-2d-118)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d-14) then
        tmp = (y * x) / (2.0d0 * a)
    else
        tmp = ((t * z) * (-9.0d0)) * (0.5d0 / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) / (2.0 * a);
	} else {
		tmp = ((t * z) * -9.0) * (0.5 / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -2e-118:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e-14:
		tmp = (y * x) / (2.0 * a)
	else:
		tmp = ((t * z) * -9.0) * (0.5 / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -2e-118)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(y * x) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(t * z) * -9.0) * Float64(0.5 / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -2e-118)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e-14)
		tmp = (y * x) / (2.0 * a);
	else
		tmp = ((t * z) * -9.0) * (0.5 / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-118], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(y * x), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * -9.0), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999997e-118

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 74.8%

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

   if -1.99999999999999997e-118 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 93.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

      1. lift-/.f64 N/A

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

      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 96.5

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

   4. Applied rewrites 96.5%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.4

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

   7. Applied rewrites 80.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t \cdot z\right) \cdot -9\right) \cdot \frac{0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a} \cdot y, x, \left(\frac{z}{a} \cdot -4.5\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= (* 2.0 a) 1e+32)
   (/ (fma (* -9.0 t) z (* y x)) (* 2.0 a))
   (fma (* (/ 0.5 a) y) x (* (* (/ z a) -4.5) t))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((2.0 * a) <= 1e+32) {
		tmp = fma((-9.0 * t), z, (y * x)) / (2.0 * a);
	} else {
		tmp = fma(((0.5 / a) * y), x, (((z / a) * -4.5) * t));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 1e+32)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(0.5 / a) * y), x, Float64(Float64(Float64(z / a) * -4.5) * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(2.0 * a), $MachinePrecision], 1e+32], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x + N[(N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 1.00000000000000005e32

   1. Initial program 89.9%

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

      1. 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 90.0

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

      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 90.0

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

   4. Applied rewrites 90.0%

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

   if 1.00000000000000005e32 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 93.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 93.8%

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

   5. Taylor expanded in a around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 93.8

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

   7. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(y \cdot \frac{0.5}{a}, x, \color{blue}{\left(\frac{z}{a} \cdot -4.5\right) \cdot t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 10^{+32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{a} \cdot y, x, \left(\frac{z}{a} \cdot -4.5\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -2e-118)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e-14) (/ (* y x) (* 2.0 a)) (/ (* (* -4.5 t) z) a)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) / (2.0 * a);
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-2d-118)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d-14) then
        tmp = (y * x) / (2.0d0 * a)
    else
        tmp = (((-4.5d0) * t) * z) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) / (2.0 * a);
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -2e-118:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e-14:
		tmp = (y * x) / (2.0 * a)
	else:
		tmp = ((-4.5 * t) * z) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -2e-118)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(y * x) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -2e-118)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e-14)
		tmp = (y * x) / (2.0 * a);
	else
		tmp = ((-4.5 * t) * z) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-118], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(y * x), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999997e-118

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 74.8%

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

   if -1.99999999999999997e-118 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 93.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 78.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y \cdot x}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -2e-118)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e-14) (* (* y x) (/ 0.5 a)) (/ (* (* -4.5 t) z) a)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) * (0.5 / a);
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-2d-118)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d-14) then
        tmp = (y * x) * (0.5d0 / a)
    else
        tmp = (((-4.5d0) * t) * z) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -2e-118) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = (y * x) * (0.5 / a);
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -2e-118:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e-14:
		tmp = (y * x) * (0.5 / a)
	else:
		tmp = ((-4.5 * t) * z) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -2e-118)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(y * x) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -2e-118)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e-14)
		tmp = (y * x) * (0.5 / a);
	else
		tmp = ((-4.5 * t) * z) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-118], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(y * x), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -1.99999999999999997e-118

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.1%

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

   9. Applied rewrites 74.8%

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

   if -1.99999999999999997e-118 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 93.2%

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

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

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

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

      22. metadata-eval 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 78.9%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -2 \cdot 10^{-118}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -1e-76)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e-14) (* (* (/ x a) 0.5) y) (/ (* (* -4.5 t) z) a)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-1d-76)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d-14) then
        tmp = ((x / a) * 0.5d0) * y
    else
        tmp = (((-4.5d0) * t) * z) / a
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 * t) * z) / a;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -1e-76:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e-14:
		tmp = ((x / a) * 0.5) * y
	else:
		tmp = ((-4.5 * t) * z) / a
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -1e-76)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	else
		tmp = Float64(Float64(Float64(-4.5 * t) * z) / a);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -1e-76)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e-14)
		tmp = ((x / a) * 0.5) * y;
	else
		tmp = ((-4.5 * t) * z) / a;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-76], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-4.5 * t), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.99999999999999927e-77

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 76.6%

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

   if -9.99999999999999927e-77 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   7. 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 72.9

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

   8. Applied rewrites 72.9%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 78.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4.5 \cdot t\right) \cdot z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -1e-76)
     (* (* -4.5 t) (/ z a))
     (if (<= t_1 2e-14) (* (* (/ x a) 0.5) y) (* (* (/ -4.5 a) t) z)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-1d-76)) then
        tmp = ((-4.5d0) * t) * (z / a)
    else if (t_1 <= 2d-14) then
        tmp = ((x / a) * 0.5d0) * y
    else
        tmp = (((-4.5d0) / a) * t) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = (-4.5 * t) * (z / a);
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -1e-76:
		tmp = (-4.5 * t) * (z / a)
	elif t_1 <= 2e-14:
		tmp = ((x / a) * 0.5) * y
	else:
		tmp = ((-4.5 / a) * t) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -1e-76)
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	else
		tmp = Float64(Float64(Float64(-4.5 / a) * t) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -1e-76)
		tmp = (-4.5 * t) * (z / a);
	elseif (t_1 <= 2e-14)
		tmp = ((x / a) * 0.5) * y;
	else
		tmp = ((-4.5 / a) * t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-76], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-4.5 / a), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.99999999999999927e-77

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 76.6%

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

   if -9.99999999999999927e-77 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   7. 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 72.9

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

   8. Applied rewrites 72.9%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -1e-76)
     (* (* (/ -4.5 a) z) t)
     (if (<= t_1 2e-14) (* (* (/ x a) 0.5) y) (* (* (/ -4.5 a) t) z)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = ((-4.5 / a) * z) * t;
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-1d-76)) then
        tmp = (((-4.5d0) / a) * z) * t
    else if (t_1 <= 2d-14) then
        tmp = ((x / a) * 0.5d0) * y
    else
        tmp = (((-4.5d0) / a) * t) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = ((-4.5 / a) * z) * t;
	} else if (t_1 <= 2e-14) {
		tmp = ((x / a) * 0.5) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -1e-76:
		tmp = ((-4.5 / a) * z) * t
	elif t_1 <= 2e-14:
		tmp = ((x / a) * 0.5) * y
	else:
		tmp = ((-4.5 / a) * t) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -1e-76)
		tmp = Float64(Float64(Float64(-4.5 / a) * z) * t);
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(x / a) * 0.5) * y);
	else
		tmp = Float64(Float64(Float64(-4.5 / a) * t) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -1e-76)
		tmp = ((-4.5 / a) * z) * t;
	elseif (t_1 <= 2e-14)
		tmp = ((x / a) * 0.5) * y;
	else
		tmp = ((-4.5 / a) * t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-76], N[(N[(N[(-4.5 / a), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[(x / a), $MachinePrecision] * 0.5), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-4.5 / a), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.99999999999999927e-77

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 76.1%

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

   if -9.99999999999999927e-77 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 92.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.1

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

   5. Applied rewrites 21.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   7. 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 72.9

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

   8. Applied rewrites 72.9%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{x}{a} \cdot 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (* 9.0 z))))
   (if (<= t_1 -1e-76)
     (* (* (/ -4.5 a) z) t)
     (if (<= t_1 2e-14) (* (* (/ 0.5 a) x) y) (* (* (/ -4.5 a) t) z)))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = ((-4.5 / a) * z) * t;
	} else if (t_1 <= 2e-14) {
		tmp = ((0.5 / a) * x) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (9.0d0 * z)
    if (t_1 <= (-1d-76)) then
        tmp = (((-4.5d0) / a) * z) * t
    else if (t_1 <= 2d-14) then
        tmp = ((0.5d0 / a) * x) * y
    else
        tmp = (((-4.5d0) / a) * t) * z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (9.0 * z);
	double tmp;
	if (t_1 <= -1e-76) {
		tmp = ((-4.5 / a) * z) * t;
	} else if (t_1 <= 2e-14) {
		tmp = ((0.5 / a) * x) * y;
	} else {
		tmp = ((-4.5 / a) * t) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = t * (9.0 * z)
	tmp = 0
	if t_1 <= -1e-76:
		tmp = ((-4.5 / a) * z) * t
	elif t_1 <= 2e-14:
		tmp = ((0.5 / a) * x) * y
	else:
		tmp = ((-4.5 / a) * t) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(9.0 * z))
	tmp = 0.0
	if (t_1 <= -1e-76)
		tmp = Float64(Float64(Float64(-4.5 / a) * z) * t);
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(Float64(0.5 / a) * x) * y);
	else
		tmp = Float64(Float64(Float64(-4.5 / a) * t) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (9.0 * z);
	tmp = 0.0;
	if (t_1 <= -1e-76)
		tmp = ((-4.5 / a) * z) * t;
	elseif (t_1 <= 2e-14)
		tmp = ((0.5 / a) * x) * y;
	else
		tmp = ((-4.5 / a) * t) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-76], N[(N[(N[(-4.5 / a), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(N[(0.5 / a), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-4.5 / a), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.99999999999999927e-77

   1. Initial program 83.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 76.9%

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

   9. Applied rewrites 76.1%

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

   if -9.99999999999999927e-77 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e-14

   1. Initial program 92.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 72.9%

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

   if 2e-14 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 96.5%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   7. Applied rewrites 74.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot \left(9 \cdot z\right) \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \mathbf{elif}\;t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.1%

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

   9. Applied rewrites 94.1%

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

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

   1. Initial program 94.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

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

Alternative 14: 92.9% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= (* t (* 9.0 z)) (- INFINITY))
   (* (* -4.5 t) (/ z a))
   (/ (fma (* -9.0 z) t (* y x)) (* 2.0 a))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 42.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.1%

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

   9. Applied rewrites 94.1%

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

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

   1. Initial program 94.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 94.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.3

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

   4. Applied rewrites 94.3%

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

4. Final simplification 94.3%

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

Alternative 15: 92.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= (* t (* 9.0 z)) (- INFINITY))
   (* (* -4.5 t) (/ z a))
   (* (fma (* t z) -9.0 (* y x)) (/ 0.5 a))))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t * (9.0 * z)) <= -((double) INFINITY)) {
		tmp = (-4.5 * t) * (z / a);
	} 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])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(t * Float64(9.0 * z)) <= Float64(-Inf))
		tmp = Float64(Float64(-4.5 * t) * Float64(z / a));
	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.
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[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $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 (*.f64 z #s(literal 9 binary64)) t) < -inf.0

   1. Initial program 42.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.0

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.1%

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

   9. Applied rewrites 94.1%

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

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

   1. Initial program 94.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 94.2

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

   4. Applied rewrites 94.2%

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

4. Final simplification 94.2%

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

Alternative 16: 51.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 (<= (* 2.0 a) 3e-180) (* (* (/ -4.5 a) t) z) (* (* (/ -4.5 a) z) t)))

C

assert(x < y && y < z && z < t && t < a);
assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((2.0 * a) <= 3e-180) {
		tmp = ((-4.5 / a) * t) * z;
	} else {
		tmp = ((-4.5 / a) * z) * t;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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 ((2.0d0 * a) <= 3d-180) then
        tmp = (((-4.5d0) / a) * t) * z
    else
        tmp = (((-4.5d0) / a) * z) * t
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
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 ((2.0 * a) <= 3e-180) {
		tmp = ((-4.5 / a) * t) * z;
	} else {
		tmp = ((-4.5 / a) * z) * t;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (2.0 * a) <= 3e-180:
		tmp = ((-4.5 / a) * t) * z
	else:
		tmp = ((-4.5 / a) * z) * t
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 3e-180)
		tmp = Float64(Float64(Float64(-4.5 / a) * t) * z);
	else
		tmp = Float64(Float64(Float64(-4.5 / a) * z) * t);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
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 ((2.0 * a) <= 3e-180)
		tmp = ((-4.5 / a) * t) * z;
	else
		tmp = ((-4.5 / a) * z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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[(2.0 * a), $MachinePrecision], 3e-180], N[(N[(N[(-4.5 / a), $MachinePrecision] * t), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(-4.5 / a), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 3.0000000000000001e-180

   1. Initial program 87.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   7. Applied rewrites 55.5%

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

   if 3.0000000000000001e-180 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 95.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 46.9

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

   5. Applied rewrites 46.9%

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

   7. Applied rewrites 46.9%

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

   9. Applied rewrites 47.3%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 3 \cdot 10^{-180}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-4.5}{a} \cdot z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in t around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 52.0

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

5. Applied rewrites 52.0%

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

7. Applied rewrites 52.0%

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

8. Final simplification 52.0%

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

Developer Target 1: 93.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024277 
(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)))