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

Percentage Accurate: 90.9% → 94.0%

Time: 10.2s

Alternatives: 8

Speedup: 0.5×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+219}:\\ \;\;\;\;y \cdot \left(-4.5 \cdot \frac{z \cdot t}{y \cdot a} + 0.5 \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + \frac{0.5}{\frac{a}{x} \cdot \frac{t}{y}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (<= t_1 -2e+219)
     (* y (+ (* -4.5 (/ (* z t) (* y a))) (* 0.5 (/ x a))))
     (if (<= t_1 2e+292)
       (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
       (* t (+ (* -4.5 (/ z a)) (/ 0.5 (* (/ a x) (/ t y)))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -2e+219) {
		tmp = y * ((-4.5 * ((z * t) / (y * a))) + (0.5 * (x / a)));
	} else if (t_1 <= 2e+292) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - ((z * 9.0d0) * t)
    if (t_1 <= (-2d+219)) then
        tmp = y * (((-4.5d0) * ((z * t) / (y * a))) + (0.5d0 * (x / a)))
    else if (t_1 <= 2d+292) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = t * (((-4.5d0) * (z / a)) + (0.5d0 / ((a / x) * (t / y))))
    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 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if (t_1 <= -2e+219) {
		tmp = y * ((-4.5 * ((z * t) / (y * a))) + (0.5 * (x / a)));
	} else if (t_1 <= 2e+292) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - ((z * 9.0) * t)
	tmp = 0
	if t_1 <= -2e+219:
		tmp = y * ((-4.5 * ((z * t) / (y * a))) + (0.5 * (x / a)))
	elif t_1 <= 2e+292:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if (t_1 <= -2e+219)
		tmp = Float64(y * Float64(Float64(-4.5 * Float64(Float64(z * t) / Float64(y * a))) + Float64(0.5 * Float64(x / a))));
	elseif (t_1 <= 2e+292)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(Float64(-4.5 * Float64(z / a)) + Float64(0.5 / Float64(Float64(a / x) * Float64(t / y)))));
	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 = (x * y) - ((z * 9.0) * t);
	tmp = 0.0;
	if (t_1 <= -2e+219)
		tmp = y * ((-4.5 * ((z * t) / (y * a))) + (0.5 * (x / a)));
	elseif (t_1 <= 2e+292)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	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[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+219], N[(y * N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[(a / x), $MachinePrecision] * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -1.99999999999999993e219

   1. Initial program 79.2%

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

   3. Taylor expanded in y around inf 90.7%

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

   if -1.99999999999999993e219 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e292

   1. Initial program 98.5%

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

      1. div-sub 97.4%

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

      2. *-commutative 97.4%

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

      3. div-sub 98.5%

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

      4. cancel-sign-sub-inv 98.5%

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

      5. *-commutative 98.5%

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

      6. fma-define 98.5%

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

      7. distribute-rgt-neg-in 98.5%

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

      8. associate-*r* 98.5%

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

      9. distribute-lft-neg-in 98.5%

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

      10. *-commutative 98.5%

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

      11. distribute-rgt-neg-in 98.5%

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

      12. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. metadata-eval 98.5%

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

      4. distribute-rgt-neg-in 98.5%

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

      5. distribute-lft-neg-in 98.5%

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

      6. fma-neg 98.5%

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

      7. associate-*l* 98.5%

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

   6. Applied egg-rr 98.5%

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

   if 2e292 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 61.5%

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

   3. Taylor expanded in t around inf 72.9%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. Step-by-step derivation

      1. clear-num 72.9%

         \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{a \cdot t}{x \cdot y}}}\right) \]

      2. un-div-inv 72.9%

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

      3. times-frac 82.7%

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

   5. Applied egg-rr 82.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2 \cdot 10^{+219}:\\ \;\;\;\;y \cdot \left(-4.5 \cdot \frac{z \cdot t}{y \cdot a} + 0.5 \cdot \frac{x}{a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + \frac{0.5}{\frac{a}{x} \cdot \frac{t}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+292}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + \frac{0.5}{\frac{a}{x} \cdot \frac{t}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+292)))
     (* t (+ (* -4.5 (/ z a)) (/ 0.5 (* (/ a x) (/ t y)))))
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+292)) {
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	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 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e+292)) {
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	} else {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (x * y) - ((z * 9.0) * t)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e+292):
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))))
	else:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+292))
		tmp = Float64(t * Float64(Float64(-4.5 * Float64(z / a)) + Float64(0.5 / Float64(Float64(a / x) * Float64(t / y)))));
	else
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	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 = (x * y) - ((z * 9.0) * t);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e+292)))
		tmp = t * ((-4.5 * (z / a)) + (0.5 / ((a / x) * (t / y))));
	else
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	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[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+292]], $MachinePrecision]], N[(t * N[(N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[(a / x), $MachinePrecision] * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0 or 2e292 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 62.7%

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

   3. Taylor expanded in t around inf 73.1%

      \[\leadsto \color{blue}{t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \frac{x \cdot y}{a \cdot t}\right)} \]
   4. Step-by-step derivation

      1. clear-num 73.1%

         \[\leadsto t \cdot \left(-4.5 \cdot \frac{z}{a} + 0.5 \cdot \color{blue}{\frac{1}{\frac{a \cdot t}{x \cdot y}}}\right) \]

      2. un-div-inv 73.1%

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

      3. times-frac 81.5%

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

   5. Applied egg-rr 81.5%

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e292

   1. Initial program 98.7%

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

      1. div-sub 97.6%

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

      2. *-commutative 97.6%

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

      3. div-sub 98.7%

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

      4. cancel-sign-sub-inv 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-define 98.7%

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

      7. distribute-rgt-neg-in 98.7%

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

      8. associate-*r* 98.6%

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

      9. distribute-lft-neg-in 98.6%

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

      10. *-commutative 98.6%

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

      11. distribute-rgt-neg-in 98.6%

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

      12. metadata-eval 98.6%

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

   3. Simplified 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*r* 98.7%

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

      3. metadata-eval 98.7%

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

      4. distribute-rgt-neg-in 98.7%

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

      5. distribute-lft-neg-in 98.7%

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

      6. fma-neg 98.7%

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

      7. associate-*l* 98.6%

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

   6. Applied egg-rr 98.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+292}\right):\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a} + \frac{0.5}{\frac{a}{x} \cdot \frac{t}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.2222222222222222}{z \cdot \frac{t}{a}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 (- INFINITY))
     (* t (/ (* z -4.5) a))
     (if (<= t_1 5e+281)
       (/ (- (* x y) t_1) (* a 2.0))
       (/ 1.0 (/ -0.2222222222222222 (* z (/ t a))))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t * ((z * -4.5) / a);
	} else if (t_1 <= 5e+281) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = 1.0 / (-0.2222222222222222 / (z * (t / a)));
	}
	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 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = t * ((z * -4.5) / a);
	} else if (t_1 <= 5e+281) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = 1.0 / (-0.2222222222222222 / (z * (t / a)));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -math.inf:
		tmp = t * ((z * -4.5) / a)
	elif t_1 <= 5e+281:
		tmp = ((x * y) - t_1) / (a * 2.0)
	else:
		tmp = 1.0 / (-0.2222222222222222 / (z * (t / a)))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(z * -4.5) / a));
	elseif (t_1 <= 5e+281)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(-0.2222222222222222 / Float64(z * Float64(t / a))));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = t * ((z * -4.5) / a);
	elseif (t_1 <= 5e+281)
		tmp = ((x * y) - t_1) / (a * 2.0);
	else
		tmp = 1.0 / (-0.2222222222222222 / (z * (t / a)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t * N[(N[(z * -4.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+281], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-0.2222222222222222 / N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.8%

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

   3. Taylor expanded in x around 0 58.8%

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

      1. associate-/l* 99.7%

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

   5. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*r/ 58.8%

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

      3. metadata-eval 58.8%

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

      4. times-frac 58.8%

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

      5. associate-*r* 58.8%

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

      6. *-un-lft-identity 58.8%

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

      7. times-frac 99.8%

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

      8. times-frac 99.5%

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

      9. metadata-eval 99.5%

         \[\leadsto \frac{t}{1} \cdot \left(\frac{z}{a} \cdot \color{blue}{-4.5}\right) \]

      10. *-commutative 99.5%

          \[\leadsto \frac{t}{1} \cdot \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \]

      11. associate-*r/ 99.8%

          \[\leadsto \frac{t}{1} \cdot \color{blue}{\frac{-4.5 \cdot z}{a}} \]

      12. times-frac 58.8%

          \[\leadsto \color{blue}{\frac{t \cdot \left(-4.5 \cdot z\right)}{1 \cdot a}} \]

      13. *-un-lft-identity 58.8%

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

      14. associate-*l/ 99.7%

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

      15. *-commutative 99.7%

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

      16. clear-num 99.6%

          \[\leadsto \left(-4.5 \cdot z\right) \cdot \color{blue}{\frac{1}{\frac{a}{t}}} \]

      17. un-div-inv 99.6%

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

      18. *-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-/r/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -inf.0 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000016e281

   1. Initial program 93.8%

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

   if 5.00000000000000016e281 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 66.4%

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

   3. Taylor expanded in x around 0 73.5%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r/ 73.5%

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

      3. metadata-eval 73.5%

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

      4. times-frac 73.5%

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

      5. associate-*r* 73.5%

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

      6. clear-num 73.5%

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

      7. associate-*r* 73.5%

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

      8. times-frac 73.5%

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

      9. metadata-eval 73.5%

         \[\leadsto \frac{1}{\frac{a}{t \cdot z} \cdot \color{blue}{-0.2222222222222222}} \]

   7. Applied egg-rr 73.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{t \cdot z} \cdot -0.2222222222222222}} \]
   8. Step-by-step derivation

      1. *-commutative 73.5%

         \[\leadsto \frac{1}{\color{blue}{-0.2222222222222222 \cdot \frac{a}{t \cdot z}}} \]

      2. clear-num 73.5%

         \[\leadsto \frac{1}{-0.2222222222222222 \cdot \color{blue}{\frac{1}{\frac{t \cdot z}{a}}}} \]

      3. clear-num 73.5%

         \[\leadsto \frac{1}{-0.2222222222222222 \cdot \frac{1}{\color{blue}{\frac{1}{\frac{a}{t \cdot z}}}}} \]

      4. un-div-inv 73.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{-0.2222222222222222}{\frac{1}{\frac{a}{t \cdot z}}}}} \]

      5. clear-num 73.5%

         \[\leadsto \frac{1}{\frac{-0.2222222222222222}{\color{blue}{\frac{t \cdot z}{a}}}} \]

      6. *-commutative 73.5%

         \[\leadsto \frac{1}{\frac{-0.2222222222222222}{\frac{\color{blue}{z \cdot t}}{a}}} \]

      7. associate-/l* 99.9%

         \[\leadsto \frac{1}{\frac{-0.2222222222222222}{\color{blue}{z \cdot \frac{t}{a}}}} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.2222222222222222}{z \cdot \frac{t}{a}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+281}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.2222222222222222}{z \cdot \frac{t}{a}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{a}\right)\\ \end{array} \end{array} \]

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 (<= (* x y) -4e+289)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) 5e+288)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (* y (* x (/ 0.5 a))))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -4e+289) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= 5e+288) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = y * (x * (0.5 / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-4d+289)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= 5d+288) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = y * (x * (0.5d0 / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -4e+289) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= 5e+288) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = y * (x * (0.5 / a));
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -4e+289:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= 5e+288:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = y * (x * (0.5 / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -4e+289)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= 5e+288)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(y * Float64(x * Float64(0.5 / a)));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -4e+289)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= 5e+288)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = y * (x * (0.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+289], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+288], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.0000000000000002e289

   1. Initial program 66.5%

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

   3. Taylor expanded in x around inf 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/l* 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

   5. Simplified 99.9%

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

   if -4.0000000000000002e289 < (*.f64 x y) < 5.0000000000000003e288

   1. Initial program 94.2%

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

      1. div-sub 92.3%

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

      2. *-commutative 92.3%

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

      3. div-sub 94.2%

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

      4. cancel-sign-sub-inv 94.2%

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

      5. *-commutative 94.2%

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

      6. fma-define 94.2%

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

      7. distribute-rgt-neg-in 94.2%

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

      8. associate-*r* 94.1%

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

      9. distribute-lft-neg-in 94.1%

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

      10. *-commutative 94.1%

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

      11. distribute-rgt-neg-in 94.1%

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

      12. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-*r* 94.2%

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

      3. metadata-eval 94.2%

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

      4. distribute-rgt-neg-in 94.2%

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

      5. distribute-lft-neg-in 94.2%

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

      6. fma-neg 94.2%

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

      7. associate-*l* 94.1%

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

   6. Applied egg-rr 94.1%

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

   if 5.0000000000000003e288 < (*.f64 x y)

   1. Initial program 63.3%

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

      1. clear-num 63.3%

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

      2. inv-pow 63.3%

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

      3. *-commutative 63.3%

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

      4. associate-/l* 63.3%

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

      5. fma-neg 68.8%

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

      6. *-commutative 68.8%

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

      7. distribute-rgt-neg-in 68.8%

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

      8. distribute-rgt-neg-in 68.8%

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

      9. metadata-eval 68.8%

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

   4. Applied egg-rr 68.8%

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

      1. unpow-1 68.8%

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

      2. associate-/r* 68.8%

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

      3. metadata-eval 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in x around inf 63.3%

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

      1. associate-*r/ 63.3%

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

      2. associate-*l/ 63.3%

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

      3. associate-*r* 89.3%

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

   9. Simplified 89.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+289}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-233}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a))))
   (if (<= y -3.5e-42)
     t_1
     (if (<= y -5.6e-233)
       (* -4.5 (/ (* z t) a))
       (if (<= y 2.05e+42) (* t (* -4.5 (/ z a))) t_1)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double tmp;
	if (y <= -3.5e-42) {
		tmp = t_1;
	} else if (y <= -5.6e-233) {
		tmp = -4.5 * ((z * t) / a);
	} else if (y <= 2.05e+42) {
		tmp = t * (-4.5 * (z / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a)
    if (y <= (-3.5d-42)) then
        tmp = t_1
    else if (y <= (-5.6d-233)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (y <= 2.05d+42) then
        tmp = t * ((-4.5d0) * (z / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double tmp;
	if (y <= -3.5e-42) {
		tmp = t_1;
	} else if (y <= -5.6e-233) {
		tmp = -4.5 * ((z * t) / a);
	} else if (y <= 2.05e+42) {
		tmp = t * (-4.5 * (z / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = x * ((y * 0.5) / a)
	tmp = 0
	if y <= -3.5e-42:
		tmp = t_1
	elif y <= -5.6e-233:
		tmp = -4.5 * ((z * t) / a)
	elif y <= 2.05e+42:
		tmp = t * (-4.5 * (z / a))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a))
	tmp = 0.0
	if (y <= -3.5e-42)
		tmp = t_1;
	elseif (y <= -5.6e-233)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif (y <= 2.05e+42)
		tmp = Float64(t * Float64(-4.5 * Float64(z / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((y * 0.5) / a);
	tmp = 0.0;
	if (y <= -3.5e-42)
		tmp = t_1;
	elseif (y <= -5.6e-233)
		tmp = -4.5 * ((z * t) / a);
	elseif (y <= 2.05e+42)
		tmp = t * (-4.5 * (z / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e-42], t$95$1, If[LessEqual[y, -5.6e-233], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+42], N[(t * N[(-4.5 * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5000000000000002e-42 or 2.05e42 < y

   1. Initial program 88.9%

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

   3. Taylor expanded in x around inf 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-/l* 72.4%

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

      3. associate-*r* 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-*r/ 72.4%

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

   5. Simplified 72.4%

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

   if -3.5000000000000002e-42 < y < -5.6000000000000002e-233

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0 79.2%

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

   if -5.6000000000000002e-233 < y < 2.05e42

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 62.7%

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

      1. *-commutative 62.7%

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

      2. associate-*r* 62.7%

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

   5. Simplified 62.7%

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

      1. associate-*r* 62.7%

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

      2. times-frac 62.8%

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

      3. associate-*r/ 68.4%

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

      4. metadata-eval 68.4%

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

      5. *-commutative 68.4%

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

      6. *-commutative 68.4%

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

      7. associate-*r* 68.3%

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

      8. *-commutative 68.3%

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

   7. Applied egg-rr 68.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-233}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \left(-4.5 \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{-234}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ (* y 0.5) a))))
   (if (<= y -7.5e-23)
     t_1
     (if (<= y -8.4e-234)
       (* -4.5 (/ (* z t) a))
       (if (<= y 2.05e+42) (* -4.5 (* t (/ z a))) t_1)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double tmp;
	if (y <= -7.5e-23) {
		tmp = t_1;
	} else if (y <= -8.4e-234) {
		tmp = -4.5 * ((z * t) / a);
	} else if (y <= 2.05e+42) {
		tmp = -4.5 * (t * (z / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * ((y * 0.5d0) / a)
    if (y <= (-7.5d-23)) then
        tmp = t_1
    else if (y <= (-8.4d-234)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (y <= 2.05d+42) then
        tmp = (-4.5d0) * (t * (z / a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x * ((y * 0.5) / a);
	double tmp;
	if (y <= -7.5e-23) {
		tmp = t_1;
	} else if (y <= -8.4e-234) {
		tmp = -4.5 * ((z * t) / a);
	} else if (y <= 2.05e+42) {
		tmp = -4.5 * (t * (z / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	t_1 = x * ((y * 0.5) / a)
	tmp = 0
	if y <= -7.5e-23:
		tmp = t_1
	elif y <= -8.4e-234:
		tmp = -4.5 * ((z * t) / a)
	elif y <= 2.05e+42:
		tmp = -4.5 * (t * (z / a))
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(Float64(y * 0.5) / a))
	tmp = 0.0
	if (y <= -7.5e-23)
		tmp = t_1;
	elseif (y <= -8.4e-234)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif (y <= 2.05e+42)
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	t_1 = x * ((y * 0.5) / a);
	tmp = 0.0;
	if (y <= -7.5e-23)
		tmp = t_1;
	elseif (y <= -8.4e-234)
		tmp = -4.5 * ((z * t) / a);
	elseif (y <= 2.05e+42)
		tmp = -4.5 * (t * (z / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-23], t$95$1, If[LessEqual[y, -8.4e-234], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+42], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999998e-23 or 2.05e42 < y

   1. Initial program 88.8%

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

   3. Taylor expanded in x around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/l* 72.7%

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

      3. associate-*r* 72.7%

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

      4. *-commutative 72.7%

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

      5. associate-*r/ 72.7%

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

   5. Simplified 72.7%

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

   if -7.4999999999999998e-23 < y < -8.39999999999999965e-234

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0 80.1%

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

   if -8.39999999999999965e-234 < y < 2.05e42

   1. Initial program 87.0%

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

   3. Taylor expanded in x around 0 62.8%

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

      1. associate-/l* 68.4%

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

   5. Simplified 68.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{-234}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= (* x y) -5e+178)
   (* (/ x a) (/ y 2.0))
   (if (<= (* x y) 5e-60) (* -4.5 (/ (* z t) a)) (* x (/ (* y 0.5) a)))))

C

assert(x < y && y < z && z < t && t < a);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+178) {
		tmp = (x / a) * (y / 2.0);
	} else if ((x * y) <= 5e-60) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = x * ((y * 0.5) / a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((x * y) <= (-5d+178)) then
        tmp = (x / a) * (y / 2.0d0)
    else if ((x * y) <= 5d-60) then
        tmp = (-4.5d0) * ((z * t) / a)
    else
        tmp = x * ((y * 0.5d0) / a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+178) {
		tmp = (x / a) * (y / 2.0);
	} else if ((x * y) <= 5e-60) {
		tmp = -4.5 * ((z * t) / a);
	} else {
		tmp = x * ((y * 0.5) / a);
	}
	return tmp;
}

Python

[x, y, z, t, a] = sort([x, y, z, t, a])
def code(x, y, z, t, a):
	tmp = 0
	if (x * y) <= -5e+178:
		tmp = (x / a) * (y / 2.0)
	elif (x * y) <= 5e-60:
		tmp = -4.5 * ((z * t) / a)
	else:
		tmp = x * ((y * 0.5) / a)
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -5e+178)
		tmp = Float64(Float64(x / a) * Float64(y / 2.0));
	elseif (Float64(x * y) <= 5e-60)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	else
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	end
	return tmp
end

MATLAB

x, y, z, t, a = num2cell(sort([x, y, z, t, a])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x * y) <= -5e+178)
		tmp = (x / a) * (y / 2.0);
	elseif ((x * y) <= 5e-60)
		tmp = -4.5 * ((z * t) / a);
	else
		tmp = x * ((y * 0.5) / a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+178], N[(N[(x / a), $MachinePrecision] * N[(y / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-60], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999999e178

   1. Initial program 80.3%

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

      1. clear-num 80.2%

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

      2. inv-pow 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/l* 80.2%

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

      5. fma-neg 80.2%

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

      6. *-commutative 80.2%

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

      7. distribute-rgt-neg-in 80.2%

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

      8. distribute-rgt-neg-in 80.2%

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

      9. metadata-eval 80.2%

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

   4. Applied egg-rr 80.2%

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

      1. unpow-1 80.2%

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

      2. associate-/r* 80.2%

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

      3. metadata-eval 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in x around inf 75.7%

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

      1. div-inv 75.7%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{\frac{a}{x \cdot y}}} \]

      2. metadata-eval 75.7%

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

      3. clear-num 75.7%

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

      4. times-frac 75.7%

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

      5. *-un-lft-identity 75.7%

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

      6. *-commutative 75.7%

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

      7. times-frac 95.4%

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

   9. Applied egg-rr 95.4%

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

   if -4.9999999999999999e178 < (*.f64 x y) < 5.0000000000000001e-60

   1. Initial program 93.2%

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

   3. Taylor expanded in x around 0 74.5%

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

   if 5.0000000000000001e-60 < (*.f64 x y)

   1. Initial program 87.9%

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

   3. Taylor expanded in x around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-/l* 73.3%

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

      3. associate-*r* 73.3%

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

      4. *-commutative 73.3%

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

      5. associate-*r/ 73.3%

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

   5. Simplified 73.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.6% accurate, 1.9× speedup

Math

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

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

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 0 52.0%

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

   1. associate-/l* 52.3%

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

5. Simplified 52.3%

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

Developer Target 1: 93.0% accurate, 0.5× 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 2024118 
(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)))