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

Percentage Accurate: 90.7% → 94.5%

Time: 11.9s

Alternatives: 13

Speedup: 0.6×

• 27.7% 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: 90.7% 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: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x \cdot y}{z}, -4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)))
   (if (<= t_1 -1e+126)
     (* (fma 0.5 (/ (* x y) z) (* -4.5 t)) (/ z a))
     (if (<= t_1 2e+303)
       (/ (- (* x y) t_1) (* 2.0 a))
       (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -1e+126) {
		tmp = fma(0.5, ((x * y) / z), (-4.5 * t)) * (z / a);
	} else if (t_1 <= 2e+303) {
		tmp = ((x * y) - t_1) / (2.0 * a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	tmp = 0.0
	if (t_1 <= -1e+126)
		tmp = Float64(fma(0.5, Float64(Float64(x * y) / z), Float64(-4.5 * t)) * Float64(z / a));
	elseif (t_1 <= 2e+303)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(t / a) * -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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+126], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] + N[(-4.5 * t), $MachinePrecision]), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -9.99999999999999925e125

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.2%

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

   if -9.99999999999999925e125 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e303

   1. Initial program 97.3%

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

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

   1. Initial program 61.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 25.0

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

   5. Applied rewrites 25.0%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x \cdot y}{z}, -4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x \cdot y - \left(9 \cdot z\right) \cdot t}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)))
   (if (<= t_1 (- INFINITY))
     (* (* -4.5 (/ z a)) t)
     (if (<= t_1 2e+303)
       (/ (- (* x y) t_1) (* 2.0 a))
       (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-4.5 * (z / a)) * t;
	} else if (t_1 <= 2e+303) {
		tmp = ((x * y) - t_1) / (2.0 * a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (-4.5 * (z / a)) * t;
	} else if (t_1 <= 2e+303) {
		tmp = ((x * y) - t_1) / (2.0 * a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (-4.5 * (z / a)) * t
	elif t_1 <= 2e+303:
		tmp = ((x * y) - t_1) / (2.0 * a)
	else:
		tmp = ((t / a) * -4.5) * z
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-4.5 * Float64(z / a)) * t);
	elseif (t_1 <= 2e+303)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(t / a) * -4.5) * z);
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (-4.5 * (z / a)) * t;
	elseif (t_1 <= 2e+303)
		tmp = ((x * y) - t_1) / (2.0 * a);
	else
		tmp = ((t / a) * -4.5) * z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.2%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e303

   1. Initial program 97.2%

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

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

   1. Initial program 61.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 25.0

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

   5. Applied rewrites 25.0%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 97.5%

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

Alternative 3: 95.0% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)))
   (if (<= t_1 (- INFINITY))
     (* (* -4.5 (/ z a)) t)
     (if (<= t_1 2e+303)
       (/ (fma (* -9.0 z) t (* x y)) (* 2.0 a))
       (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-4.5 * (z / a)) * t;
	} else if (t_1 <= 2e+303) {
		tmp = fma((-9.0 * z), t, (x * y)) / (2.0 * a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-4.5 * Float64(z / a)) * t);
	elseif (t_1 <= 2e+303)
		tmp = Float64(fma(Float64(-9.0 * z), t, Float64(x * y)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(t / a) * -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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(N[(-9.0 * z), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.2%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e303

   1. Initial program 97.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 97.2

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

      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 97.2

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

   4. Applied rewrites 97.2%

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

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

   1. Initial program 61.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 25.0

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

   5. Applied rewrites 25.0%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 97.5%

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

Alternative 4: 95.0% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)))
   (if (<= t_1 (- INFINITY))
     (* (* -4.5 (/ z a)) t)
     (if (<= t_1 2e+303)
       (/ (fma y x (* -9.0 (* z t))) (* 2.0 a))
       (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-4.5 * (z / a)) * t;
	} else if (t_1 <= 2e+303) {
		tmp = fma(y, x, (-9.0 * (z * t))) / (2.0 * a);
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-4.5 * Float64(z / a)) * t);
	elseif (t_1 <= 2e+303)
		tmp = Float64(fma(y, x, Float64(-9.0 * Float64(z * t))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(t / a) * -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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(y * x + N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.2%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e303

   1. Initial program 97.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 97.2

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

   4. Applied rewrites 97.2%

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

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

   1. Initial program 61.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 25.0

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

   5. Applied rewrites 25.0%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 97.5%

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

Alternative 5: 94.9% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)))
   (if (<= t_1 (- INFINITY))
     (* (* -4.5 (/ z a)) t)
     (if (<= t_1 2e+303)
       (* (/ 0.5 a) (fma (* z t) -9.0 (* x y)))
       (* (* (/ t a) -4.5) z)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-4.5 * (z / a)) * t;
	} else if (t_1 <= 2e+303) {
		tmp = (0.5 / a) * fma((z * t), -9.0, (x * y));
	} else {
		tmp = ((t / a) * -4.5) * z;
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-4.5 * Float64(z / a)) * t);
	elseif (t_1 <= 2e+303)
		tmp = Float64(Float64(0.5 / a) * fma(Float64(z * t), -9.0, Float64(x * y)));
	else
		tmp = Float64(Float64(Float64(t / a) * -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.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], N[(N[(0.5 / a), $MachinePrecision] * N[(N[(z * t), $MachinePrecision] * -9.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.2%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 2e303

   1. Initial program 97.2%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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 97.1

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

   4. Applied rewrites 97.1%

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

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

   1. Initial program 61.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   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. 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 25.0

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

   5. Applied rewrites 25.0%

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

4. Final simplification 97.4%

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

Alternative 6: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, -9 \cdot \left(z \cdot t\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{y}{a}, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* 2.0 a) 4e-75)
   (/ (fma y x (* -9.0 (* z t))) (* 2.0 a))
   (fma (* 0.5 x) (/ y a) (* (* -4.5 (/ z a)) t))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((2.0 * a) <= 4e-75) {
		tmp = fma(y, x, (-9.0 * (z * t))) / (2.0 * a);
	} else {
		tmp = fma((0.5 * x), (y / a), ((-4.5 * (z / a)) * t));
	}
	return tmp;
}

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(2.0 * a) <= 4e-75)
		tmp = Float64(fma(y, x, Float64(-9.0 * Float64(z * t))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(0.5 * x), Float64(y / a), Float64(Float64(-4.5 * Float64(z / a)) * t));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(2.0 * a), $MachinePrecision], 4e-75], N[(N[(y * x + N[(-9.0 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(y / a), $MachinePrecision] + N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a #s(literal 2 binary64)) < 3.9999999999999998e-75

   1. Initial program 95.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 95.5

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

   4. Applied rewrites 95.5%

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

   if 3.9999999999999998e-75 < (*.f64 a #s(literal 2 binary64))

   1. Initial program 87.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. associate-/l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r/ N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. associate-*l/ N/A

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

      22. associate-*r/ N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 96.9%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot a \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, -9 \cdot \left(z \cdot t\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{y}{a}, \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ t_2 := \left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)) (t_2 (* (* -4.5 t) (/ z a))))
   (if (<= t_1 -5e-23)
     t_2
     (if (<= t_1 5000000000000.0) (/ (* x y) (* 2.0 a)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * t) * (z / a);
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = (x * y) / (2.0 * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * z) * t
    t_2 = ((-4.5d0) * t) * (z / a)
    if (t_1 <= (-5d-23)) then
        tmp = t_2
    else if (t_1 <= 5000000000000.0d0) then
        tmp = (x * y) / (2.0d0 * a)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * t) * (z / a);
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = (x * y) / (2.0 * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	t_2 = (-4.5 * t) * (z / a)
	tmp = 0
	if t_1 <= -5e-23:
		tmp = t_2
	elif t_1 <= 5000000000000.0:
		tmp = (x * y) / (2.0 * a)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	t_2 = Float64(Float64(-4.5 * t) * Float64(z / a))
	tmp = 0.0
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		tmp = Float64(Float64(x * y) / Float64(2.0 * a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	t_2 = (-4.5 * t) * (z / a);
	tmp = 0.0;
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		tmp = (x * y) / (2.0 * a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-23], t$95$2, If[LessEqual[t$95$1, 5000000000000.0], N[(N[(x * y), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000002e-23 or 5e12 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.6

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

   10. Applied rewrites 74.6%

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

   12. Applied rewrites 77.3%

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

   if -5.0000000000000002e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5e12

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 5000000000000:\\ \;\;\;\;\frac{x \cdot y}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ t_2 := \left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5000000000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)) (t_2 (* (* -4.5 t) (/ z a))))
   (if (<= t_1 -5e-23)
     t_2
     (if (<= t_1 5000000000000.0) (* (/ 0.5 a) (* x y)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * t) * (z / a);
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * z) * t
    t_2 = ((-4.5d0) * t) * (z / a)
    if (t_1 <= (-5d-23)) then
        tmp = t_2
    else if (t_1 <= 5000000000000.0d0) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * t) * (z / a);
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 5000000000000.0) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	t_2 = (-4.5 * t) * (z / a)
	tmp = 0
	if t_1 <= -5e-23:
		tmp = t_2
	elif t_1 <= 5000000000000.0:
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	t_2 = Float64(Float64(-4.5 * t) * Float64(z / a))
	tmp = 0.0
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	t_2 = (-4.5 * t) * (z / a);
	tmp = 0.0;
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 5000000000000.0)
		tmp = (0.5 / a) * (x * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.5 * t), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-23], t$95$2, If[LessEqual[t$95$1, 5000000000000.0], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000002e-23 or 5e12 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 28.0

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

   5. Applied rewrites 28.0%

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 74.6

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

   10. Applied rewrites 74.6%

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

   12. Applied rewrites 77.3%

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

   if -5.0000000000000002e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5e12

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   7. Applied rewrites 86.7%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 5000000000000:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot t\right) \cdot \frac{z}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ t_2 := \left(-4.5 \cdot \frac{z}{a}\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)) (t_2 (* (* -4.5 (/ z a)) t)))
   (if (<= t_1 -5e-23) t_2 (if (<= t_1 2e+39) (* (/ 0.5 a) (* x y)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * (z / a)) * t;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * z) * t
    t_2 = ((-4.5d0) * (z / a)) * t
    if (t_1 <= (-5d-23)) then
        tmp = t_2
    else if (t_1 <= 2d+39) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = (-4.5 * (z / a)) * t;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	t_2 = (-4.5 * (z / a)) * t
	tmp = 0
	if t_1 <= -5e-23:
		tmp = t_2
	elif t_1 <= 2e+39:
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	t_2 = Float64(Float64(-4.5 * Float64(z / a)) * t)
	tmp = 0.0
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	t_2 = (-4.5 * (z / a)) * t;
	tmp = 0.0;
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = (0.5 / a) * (x * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-23], t$95$2, If[LessEqual[t$95$1, 2e+39], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000002e-23 or 1.99999999999999988e39 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if -5.0000000000000002e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999988e39

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.1%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left(-4.5 \cdot \frac{z}{a}\right) \cdot t\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-4.5 \cdot \frac{z}{a}\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ t_2 := \left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)) (t_2 (* (* (/ t a) -4.5) z)))
   (if (<= t_1 -5e-23) t_2 (if (<= t_1 2e+39) (* (/ 0.5 a) (* x y)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = ((t / a) * -4.5) * z;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * z) * t
    t_2 = ((t / a) * (-4.5d0)) * z
    if (t_1 <= (-5d-23)) then
        tmp = t_2
    else if (t_1 <= 2d+39) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = ((t / a) * -4.5) * z;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	t_2 = ((t / a) * -4.5) * z
	tmp = 0
	if t_1 <= -5e-23:
		tmp = t_2
	elif t_1 <= 2e+39:
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	t_2 = Float64(Float64(Float64(t / a) * -4.5) * z)
	tmp = 0.0
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	t_2 = ((t / a) * -4.5) * z;
	tmp = 0.0;
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = (0.5 / a) * (x * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t / a), $MachinePrecision] * -4.5), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-23], t$95$2, If[LessEqual[t$95$1, 2e+39], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000002e-23 or 1.99999999999999988e39 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.7

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

   10. Applied rewrites 75.7%

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

   if -5.0000000000000002e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999988e39

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.1%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot -4.5\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(9 \cdot z\right) \cdot t\\ t_2 := \left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-23}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* 9.0 z) t)) (t_2 (* (* (/ t a) z) -4.5)))
   (if (<= t_1 -5e-23) t_2 (if (<= t_1 2e+39) (* (/ 0.5 a) (* x y)) t_2))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = ((t / a) * z) * -4.5;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (9.0d0 * z) * t
    t_2 = ((t / a) * z) * (-4.5d0)
    if (t_1 <= (-5d-23)) then
        tmp = t_2
    else if (t_1 <= 2d+39) then
        tmp = (0.5d0 / a) * (x * y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (9.0 * z) * t;
	double t_2 = ((t / a) * z) * -4.5;
	double tmp;
	if (t_1 <= -5e-23) {
		tmp = t_2;
	} else if (t_1 <= 2e+39) {
		tmp = (0.5 / a) * (x * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (9.0 * z) * t
	t_2 = ((t / a) * z) * -4.5
	tmp = 0
	if t_1 <= -5e-23:
		tmp = t_2
	elif t_1 <= 2e+39:
		tmp = (0.5 / a) * (x * y)
	else:
		tmp = t_2
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(9.0 * z) * t)
	t_2 = Float64(Float64(Float64(t / a) * z) * -4.5)
	tmp = 0.0
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = Float64(Float64(0.5 / a) * Float64(x * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (9.0 * z) * t;
	t_2 = ((t / a) * z) * -4.5;
	tmp = 0.0;
	if (t_1 <= -5e-23)
		tmp = t_2;
	elseif (t_1 <= 2e+39)
		tmp = (0.5 / a) * (x * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(9.0 * z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t / a), $MachinePrecision] * z), $MachinePrecision] * -4.5), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-23], t$95$2, If[LessEqual[t$95$1, 2e+39], N[(N[(0.5 / a), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -5.0000000000000002e-23 or 1.99999999999999988e39 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 89.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 27.7

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

   5. Applied rewrites 27.7%

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 75.7

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

   10. Applied rewrites 75.7%

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

   12. Applied rewrites 75.7%

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

   if -5.0000000000000002e-23 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 1.99999999999999988e39

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 86.1%

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{0.5}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(9 \cdot z\right) \cdot t \leq -5 \cdot 10^{-23}:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \mathbf{elif}\;\left(9 \cdot z\right) \cdot t \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t}{a} \cdot z\right) \cdot -4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.8% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* (/ 0.5 a) (* x y)))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	return (0.5 / a) * (x * y);
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (0.5d0 / a) * (x * y)
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	return (0.5 / a) * (x * y);
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	return (0.5 / a) * (x * y)

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	return Float64(Float64(0.5 / a) * Float64(x * y))
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp = code(x, y, z, t, a)
	tmp = (0.5 / a) * (x * y);
end

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 53.5

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

5. Applied rewrites 53.5%

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

7. Applied rewrites 55.5%

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

8. Final simplification 55.5%

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

Alternative 13: 51.9% accurate, 1.6× speedup

Math

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

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (* (* (/ 0.5 a) y) x))

C

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.
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;
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])
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])
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])){:}
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.
code[x_, y_, z_, t_, a_] := N[(N[(N[(0.5 / a), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in x around inf

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

   1. associate-*l/ N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 53.5

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

5. Applied rewrites 53.5%

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

7. Applied rewrites 55.8%

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

8. Final simplification 55.8%

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

Developer Target 1: 93.2% 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 2024235 
(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)))