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

Percentage Accurate: 91.3% → 96.3%

Time: 10.6s

Alternatives: 9

Speedup: 0.7×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% 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: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := y \cdot x - t \cdot \left(9 \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a + a}, x, \frac{t}{a} \cdot \left(-4.5 \cdot z\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot 0.5, y, -4.5 \cdot t\right)}{a} \cdot z\\ \end{array} \end{array} \]

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 (- (* y x) (* t (* 9.0 z)))))
   (if (<= t_1 -5e+158)
     (fma (/ y (+ a a)) x (* (/ t a) (* -4.5 z)))
     (if (<= t_1 2e+303)
       (/ (fma (* -9.0 t) z (* y x)) (+ a a))
       (* (/ (fma (* (/ x z) 0.5) y (* -4.5 t)) a) 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 = (y * x) - (t * (9.0 * z));
	double tmp;
	if (t_1 <= -5e+158) {
		tmp = fma((y / (a + a)), x, ((t / a) * (-4.5 * z)));
	} else if (t_1 <= 2e+303) {
		tmp = fma((-9.0 * t), z, (y * x)) / (a + a);
	} else {
		tmp = (fma(((x / z) * 0.5), y, (-4.5 * t)) / a) * 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(Float64(y * x) - Float64(t * Float64(9.0 * z)))
	tmp = 0.0
	if (t_1 <= -5e+158)
		tmp = fma(Float64(y / Float64(a + a)), x, Float64(Float64(t / a) * Float64(-4.5 * z)));
	elseif (t_1 <= 2e+303)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(a + a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(x / z) * 0.5), y, Float64(-4.5 * t)) / a) * z);
	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, -5e+158], N[(N[(y / N[(a + a), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t / a), $MachinePrecision] * N[(-4.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / z), $MachinePrecision] * 0.5), $MachinePrecision] * y + N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -4.9999999999999996e158

   1. Initial program 77.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 96.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 94.9

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

   8. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 94.9

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

   10. Applied rewrites 94.9%

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

   if -4.9999999999999996e158 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e303

   1. Initial program 98.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 98.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 98.4

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

   6. Applied rewrites 98.4%

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

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

   1. Initial program 70.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. div-add-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 86.3

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

   7. Applied rewrites 86.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a + a}, x, \frac{t}{a} \cdot \left(-4.5 \cdot z\right)\right)\\ \mathbf{elif}\;y \cdot x - t \cdot \left(9 \cdot z\right) \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z} \cdot 0.5, y, -4.5 \cdot t\right)}{a} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := t \cdot \left(9 \cdot z\right)\\ t_2 := \left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot t, z, y \cdot x\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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))) (t_2 (* (* (/ z a) -4.5) t)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 5e+265) (/ (fma (* -9.0 t) z (* y x)) (+ a a)) t_2))))

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 t_2 = ((z / a) * -4.5) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 5e+265) {
		tmp = fma((-9.0 * t), z, (y * x)) / (a + a);
	} else {
		tmp = t_2;
	}
	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))
	t_2 = Float64(Float64(Float64(z / a) * -4.5) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 5e+265)
		tmp = Float64(fma(Float64(-9.0 * t), z, Float64(y * x)) / Float64(a + a));
	else
		tmp = t_2;
	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[(t * N[(9.0 * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 5e+265], N[(N[(N[(-9.0 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -inf.0 or 5.0000000000000002e265 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 95.6%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000002e265

   1. Initial program 93.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 93.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 93.5

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

   6. Applied rewrites 93.5%

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

4. Final simplification 93.9%

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

Alternative 3: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(-9 \cdot z\right) \cdot t}{a + a}\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (/ 0.5 a) y) x)))
   (if (<= (* y x) -1e-12)
     t_1
     (if (<= (* y x) -5e-253)
       (/ (* (* -9.0 z) t) (+ a a))
       (if (<= (* y x) 50000.0) (* (* (/ t a) z) -4.5) 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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= -5e-253) {
		tmp = ((-9.0 * z) * t) / (a + a);
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5d0 / a) * y) * x
    if ((y * x) <= (-1d-12)) then
        tmp = t_1
    else if ((y * x) <= (-5d-253)) then
        tmp = (((-9.0d0) * z) * t) / (a + a)
    else if ((y * x) <= 50000.0d0) then
        tmp = ((t / a) * z) * (-4.5d0)
    else
        tmp = t_1
    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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= -5e-253) {
		tmp = ((-9.0 * z) * t) / (a + a);
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5 / a) * y) * x
	tmp = 0
	if (y * x) <= -1e-12:
		tmp = t_1
	elif (y * x) <= -5e-253:
		tmp = ((-9.0 * z) * t) / (a + a)
	elif (y * x) <= 50000.0:
		tmp = ((t / a) * z) * -4.5
	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 = Float64(Float64(Float64(0.5 / a) * y) * x)
	tmp = 0.0
	if (Float64(y * x) <= -1e-12)
		tmp = t_1;
	elseif (Float64(y * x) <= -5e-253)
		tmp = Float64(Float64(Float64(-9.0 * z) * t) / Float64(a + a));
	elseif (Float64(y * x) <= 50000.0)
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	else
		tmp = t_1;
	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 = ((0.5 / a) * y) * x;
	tmp = 0.0;
	if ((y * x) <= -1e-12)
		tmp = t_1;
	elseif ((y * x) <= -5e-253)
		tmp = ((-9.0 * z) * t) / (a + a);
	elseif ((y * x) <= 50000.0)
		tmp = ((t / a) * z) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e-12], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], -5e-253], N[(N[(N[(-9.0 * z), $MachinePrecision] * t), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 50000.0], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999998e-13 or 5e4 < (*.f64 x y)

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 77.2%

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

   if -9.9999999999999998e-13 < (*.f64 x y) < -4.99999999999999971e-253

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 77.1

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 77.1%

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

   if -4.99999999999999971e-253 < (*.f64 x y) < 5e4

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 78.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{\left(-9 \cdot z\right) \cdot t}{a + a}\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{t \cdot z}{a} \cdot -4.5\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (/ 0.5 a) y) x)))
   (if (<= (* y x) -1e-12)
     t_1
     (if (<= (* y x) -5e-253)
       (* (/ (* t z) a) -4.5)
       (if (<= (* y x) 50000.0) (* (* (/ t a) z) -4.5) 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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= -5e-253) {
		tmp = ((t * z) / a) * -4.5;
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5d0 / a) * y) * x
    if ((y * x) <= (-1d-12)) then
        tmp = t_1
    else if ((y * x) <= (-5d-253)) then
        tmp = ((t * z) / a) * (-4.5d0)
    else if ((y * x) <= 50000.0d0) then
        tmp = ((t / a) * z) * (-4.5d0)
    else
        tmp = t_1
    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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= -5e-253) {
		tmp = ((t * z) / a) * -4.5;
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5 / a) * y) * x
	tmp = 0
	if (y * x) <= -1e-12:
		tmp = t_1
	elif (y * x) <= -5e-253:
		tmp = ((t * z) / a) * -4.5
	elif (y * x) <= 50000.0:
		tmp = ((t / a) * z) * -4.5
	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 = Float64(Float64(Float64(0.5 / a) * y) * x)
	tmp = 0.0
	if (Float64(y * x) <= -1e-12)
		tmp = t_1;
	elseif (Float64(y * x) <= -5e-253)
		tmp = Float64(Float64(Float64(t * z) / a) * -4.5);
	elseif (Float64(y * x) <= 50000.0)
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	else
		tmp = t_1;
	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 = ((0.5 / a) * y) * x;
	tmp = 0.0;
	if ((y * x) <= -1e-12)
		tmp = t_1;
	elseif ((y * x) <= -5e-253)
		tmp = ((t * z) / a) * -4.5;
	elseif ((y * x) <= 50000.0)
		tmp = ((t / a) * z) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e-12], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], -5e-253], N[(N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision] * -4.5), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 50000.0], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -9.9999999999999998e-13 or 5e4 < (*.f64 x y)

   1. Initial program 86.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 77.2%

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

   if -9.9999999999999998e-13 < (*.f64 x y) < -4.99999999999999971e-253

   1. Initial program 97.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   if -4.99999999999999971e-253 < (*.f64 x y) < 5e4

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 78.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq -5 \cdot 10^{-253}:\\ \;\;\;\;\frac{t \cdot z}{a} \cdot -4.5\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a + a}, x, \frac{t}{a} \cdot \left(-4.5 \cdot z\right)\right)\\ \end{array} \end{array} \]

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) 4e+36)
   (/ (fma (* -9.0 z) t (* y x)) (* 2.0 a))
   (fma (/ y (+ a a)) x (* (/ t a) (* -4.5 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 tmp;
	if ((2.0 * a) <= 4e+36) {
		tmp = fma((-9.0 * z), t, (y * x)) / (2.0 * a);
	} else {
		tmp = fma((y / (a + a)), x, ((t / a) * (-4.5 * 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)
	tmp = 0.0
	if (Float64(2.0 * a) <= 4e+36)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(y * x)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(y / Float64(a + a)), x, Float64(Float64(t / a) * Float64(-4.5 * z)));
	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], 4e+36], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a + a), $MachinePrecision]), $MachinePrecision] * x + N[(N[(t / a), $MachinePrecision] * N[(-4.5 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 4.00000000000000017e36

   1. Initial program 91.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 91.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

   if 4.00000000000000017e36 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

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

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

   4. Applied rewrites 89.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 89.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 89.6

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

   6. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 94.6

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

   8. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 94.6

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

   10. Applied rewrites 94.6%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-9 \cdot z, t, y \cdot x\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a + a}, x, \frac{t}{a} \cdot \left(-4.5 \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (/ 0.5 a) y) x)))
   (if (<= (* y x) -2e-22)
     t_1
     (if (<= (* y x) 50000.0) (* (* (/ t a) z) -4.5) 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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -2e-22) {
		tmp = t_1;
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5d0 / a) * y) * x
    if ((y * x) <= (-2d-22)) then
        tmp = t_1
    else if ((y * x) <= 50000.0d0) then
        tmp = ((t / a) * z) * (-4.5d0)
    else
        tmp = t_1
    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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -2e-22) {
		tmp = t_1;
	} else if ((y * x) <= 50000.0) {
		tmp = ((t / a) * z) * -4.5;
	} else {
		tmp = t_1;
	}
	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 = ((0.5 / a) * y) * x
	tmp = 0
	if (y * x) <= -2e-22:
		tmp = t_1
	elif (y * x) <= 50000.0:
		tmp = ((t / a) * z) * -4.5
	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 = Float64(Float64(Float64(0.5 / a) * y) * x)
	tmp = 0.0
	if (Float64(y * x) <= -2e-22)
		tmp = t_1;
	elseif (Float64(y * x) <= 50000.0)
		tmp = Float64(Float64(Float64(t / a) * z) * -4.5);
	else
		tmp = t_1;
	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 = ((0.5 / a) * y) * x;
	tmp = 0.0;
	if ((y * x) <= -2e-22)
		tmp = t_1;
	elseif ((y * x) <= 50000.0)
		tmp = ((t / a) * z) * -4.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -2e-22], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 50000.0], N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000001e-22 or 5e4 < (*.f64 x y)

   1. Initial program 86.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 76.4%

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

   if -2.0000000000000001e-22 < (*.f64 x y) < 5e4

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq 50000:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot x \leq 5000:\\ \;\;\;\;\left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* (/ 0.5 a) y) x)))
   (if (<= (* y x) -1e-12)
     t_1
     (if (<= (* y x) 5000.0) (* (* (/ z a) -4.5) t) 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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= 5000.0) {
		tmp = ((z / a) * -4.5) * t;
	} else {
		tmp = t_1;
	}
	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 = ((0.5d0 / a) * y) * x
    if ((y * x) <= (-1d-12)) then
        tmp = t_1
    else if ((y * x) <= 5000.0d0) then
        tmp = ((z / a) * (-4.5d0)) * t
    else
        tmp = t_1
    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 = ((0.5 / a) * y) * x;
	double tmp;
	if ((y * x) <= -1e-12) {
		tmp = t_1;
	} else if ((y * x) <= 5000.0) {
		tmp = ((z / a) * -4.5) * t;
	} else {
		tmp = t_1;
	}
	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 = ((0.5 / a) * y) * x
	tmp = 0
	if (y * x) <= -1e-12:
		tmp = t_1
	elif (y * x) <= 5000.0:
		tmp = ((z / a) * -4.5) * t
	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 = Float64(Float64(Float64(0.5 / a) * y) * x)
	tmp = 0.0
	if (Float64(y * x) <= -1e-12)
		tmp = t_1;
	elseif (Float64(y * x) <= 5000.0)
		tmp = Float64(Float64(Float64(z / a) * -4.5) * t);
	else
		tmp = t_1;
	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 = ((0.5 / a) * y) * x;
	tmp = 0.0;
	if ((y * x) <= -1e-12)
		tmp = t_1;
	elseif ((y * x) <= 5000.0)
		tmp = ((z / a) * -4.5) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -1e-12], t$95$1, If[LessEqual[N[(y * x), $MachinePrecision], 5000.0], N[(N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.9999999999999998e-13 or 5e3 < (*.f64 x y)

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   7. Applied rewrites 76.6%

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

   if -9.9999999999999998e-13 < (*.f64 x y) < 5e3

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 76.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \mathbf{elif}\;y \cdot x \leq 5000:\\ \;\;\;\;\left(\frac{z}{a} \cdot -4.5\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{a} \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\\\ [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \left(\frac{0.5}{a} \cdot y\right) \cdot x \end{array} \]

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 (* (* (/ 0.5 a) y) x))

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 ((0.5 / a) * y) * x;
}

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 = ((0.5d0 / a) * y) * x
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 ((0.5 / a) * y) * x;
}

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 ((0.5 / a) * y) * x

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(0.5 / a) * y) * x)
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 = ((0.5 / a) * y) * x;
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[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 48.7

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

5. Applied rewrites 48.7%

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

7. Applied rewrites 49.9%

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

8. Final simplification 49.9%

   \[\leadsto \left(\frac{0.5}{a} \cdot y\right) \cdot x \]
9. Add Preprocessing

Alternative 9: 50.6% accurate, 1.8× speedup

Math

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

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 (/ (* y x) (+ a 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) {
	return (y * x) / (a + a);
}

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 = (y * x) / (a + a)
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 (y * x) / (a + a);
}

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 (y * x) / (a + a)

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(y * x) / Float64(a + a))
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 = (y * x) / (a + a);
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[(y * x), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 51.8

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

5. Applied rewrites 51.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 51.8

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

7. Applied rewrites 51.8%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 48.7

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

10. Applied rewrites 48.7%

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

11. Final simplification 48.7%

    \[\leadsto \frac{y \cdot x}{a + a} \]
12. Add Preprocessing

Developer Target 1: 93.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024297 
(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)))