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

Percentage Accurate: 91.1% → 94.4%

Time: 8.5s

Alternatives: 9

Speedup: 0.4×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+271}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{t}{\frac{a}{z}} \cdot -4.5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{a \cdot -0.2222222222222222}\\ \end{array} \end{array} \]

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 -6e+271)
     (+ (* (* x y) (/ 0.5 a)) (* (/ t (/ a z)) -4.5))
     (if (<= t_1 5e+285)
       (/ (- (* x y) t_1) (* a 2.0))
       (* t (/ z (* a -0.2222222222222222)))))))

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 <= -6e+271) {
		tmp = ((x * y) * (0.5 / a)) + ((t / (a / z)) * -4.5);
	} else if (t_1 <= 5e+285) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = t * (z / (a * -0.2222222222222222));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-6d+271)) then
        tmp = ((x * y) * (0.5d0 / a)) + ((t / (a / z)) * (-4.5d0))
    else if (t_1 <= 5d+285) then
        tmp = ((x * y) - t_1) / (a * 2.0d0)
    else
        tmp = t * (z / (a * (-0.2222222222222222d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+271) {
		tmp = ((x * y) * (0.5 / a)) + ((t / (a / z)) * -4.5);
	} else if (t_1 <= 5e+285) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = t * (z / (a * -0.2222222222222222));
	}
	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 <= -6e+271:
		tmp = ((x * y) * (0.5 / a)) + ((t / (a / z)) * -4.5)
	elif t_1 <= 5e+285:
		tmp = ((x * y) - t_1) / (a * 2.0)
	else:
		tmp = t * (z / (a * -0.2222222222222222))
	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 <= -6e+271)
		tmp = Float64(Float64(Float64(x * y) * Float64(0.5 / a)) + Float64(Float64(t / Float64(a / z)) * -4.5));
	elseif (t_1 <= 5e+285)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(z / Float64(a * -0.2222222222222222)));
	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 <= -6e+271)
		tmp = ((x * y) * (0.5 / a)) + ((t / (a / z)) * -4.5);
	elseif (t_1 <= 5e+285)
		tmp = ((x * y) - t_1) / (a * 2.0);
	else
		tmp = t * (z / (a * -0.2222222222222222));
	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, -6e+271], N[(N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+285], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(z / N[(a * -0.2222222222222222), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -6.0000000000000001e271

   1. Initial program 58.1%

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

      1. div-inv 58.2%

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

      2. fma-neg 58.2%

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

      3. *-commutative 58.2%

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

      4. distribute-rgt-neg-in 58.2%

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

      5. distribute-rgt-neg-in 58.2%

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

      6. metadata-eval 58.2%

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

      7. *-commutative 58.2%

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

      8. associate-/r* 58.2%

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

      9. metadata-eval 58.2%

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

   4. Applied egg-rr 58.2%

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

      1. *-commutative 58.2%

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

      2. fma-undefine 58.2%

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

      3. distribute-rgt-in 58.2%

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

      4. clear-num 58.2%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(t \cdot \left(z \cdot -9\right)\right) \cdot \color{blue}{\frac{1}{\frac{a}{0.5}}} \]

      5. div-inv 58.2%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(t \cdot \left(z \cdot -9\right)\right) \cdot \frac{1}{\color{blue}{a \cdot \frac{1}{0.5}}} \]

      6. metadata-eval 58.2%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(t \cdot \left(z \cdot -9\right)\right) \cdot \frac{1}{a \cdot \color{blue}{2}} \]

      7. div-inv 58.1%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \color{blue}{\frac{t \cdot \left(z \cdot -9\right)}{a \cdot 2}} \]

      8. associate-*r* 58.1%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{\color{blue}{\left(t \cdot z\right) \cdot -9}}{a \cdot 2} \]

      9. times-frac 58.1%

         \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \color{blue}{\frac{t \cdot z}{a} \cdot \frac{-9}{2}} \]

      10. associate-*r/ 93.9%

          \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \color{blue}{\left(t \cdot \frac{z}{a}\right)} \cdot \frac{-9}{2} \]

      11. clear-num 93.7%

          \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(t \cdot \color{blue}{\frac{1}{\frac{a}{z}}}\right) \cdot \frac{-9}{2} \]

      12. un-div-inv 93.9%

          \[\leadsto \left(x \cdot y\right) \cdot \frac{0.5}{a} + \color{blue}{\frac{t}{\frac{a}{z}}} \cdot \frac{-9}{2} \]

      13. metadata-eval 93.9%

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

   6. Applied egg-rr 93.9%

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

   if -6.0000000000000001e271 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000016e285

   1. Initial program 96.7%

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

   if 5.00000000000000016e285 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. associate-*r* 67.3%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. times-frac 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/r/ 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -6 \cdot 10^{+271}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a} + \frac{t}{\frac{a}{z}} \cdot -4.5\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{a \cdot -0.2222222222222222}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \frac{x}{a}}{2}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\frac{\frac{z \cdot t}{-0.2222222222222222}}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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) -2e+46)
   (/ (* y (/ x a)) 2.0)
   (if (<= (* x y) -0.005)
     (/ (/ (* z t) -0.2222222222222222) a)
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (/ (* x y) (* a 2.0))
       (* -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) {
	double tmp;
	if ((x * y) <= -2e+46) {
		tmp = (y * (x / a)) / 2.0;
	} else if ((x * y) <= -0.005) {
		tmp = ((z * t) / -0.2222222222222222) / a;
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) <= (-2d+46)) then
        tmp = (y * (x / a)) / 2.0d0
    else if ((x * y) <= (-0.005d0)) then
        tmp = ((z * t) / (-0.2222222222222222d0)) / a
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) / (a * 2.0d0)
    else
        tmp = (-4.5d0) * (t * (z / 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) <= -2e+46) {
		tmp = (y * (x / a)) / 2.0;
	} else if ((x * y) <= -0.005) {
		tmp = ((z * t) / -0.2222222222222222) / a;
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / 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) <= -2e+46:
		tmp = (y * (x / a)) / 2.0
	elif (x * y) <= -0.005:
		tmp = ((z * t) / -0.2222222222222222) / a
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) / (a * 2.0)
	else:
		tmp = -4.5 * (t * (z / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+46)
		tmp = Float64(Float64(y * Float64(x / a)) / 2.0);
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(Float64(Float64(z * t) / -0.2222222222222222) / a);
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / 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) <= -2e+46)
		tmp = (y * (x / a)) / 2.0;
	elseif ((x * y) <= -0.005)
		tmp = ((z * t) / -0.2222222222222222) / a;
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) / (a * 2.0);
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+46], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(N[(N[(z * t), $MachinePrecision] / -0.2222222222222222), $MachinePrecision] / a), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e46

   1. Initial program 85.3%

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

   3. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/l* 75.7%

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

      3. associate-*r* 75.7%

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

      4. *-commutative 75.7%

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

      5. associate-*r/ 75.7%

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

   5. Simplified 75.7%

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

      1. associate-*r/ 69.5%

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

      2. associate-*l/ 77.6%

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

      3. *-commutative 77.6%

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

      4. metadata-eval 77.6%

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

      5. div-inv 77.6%

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

      6. associate-*r/ 77.6%

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

   7. Applied egg-rr 77.6%

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

   if -2e46 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 60.4%

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

      1. associate-*r/ 60.6%

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

      2. associate-*r* 60.7%

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

      3. associate-*l/ 52.3%

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

      4. associate-*r/ 52.3%

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

      5. *-commutative 52.3%

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

      6. associate-*r/ 52.3%

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

   5. Simplified 52.3%

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

      1. clear-num 52.1%

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

      2. un-div-inv 53.4%

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

      3. *-un-lft-identity 53.4%

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

      4. times-frac 53.8%

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

      5. metadata-eval 53.8%

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

   7. Applied egg-rr 53.8%

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

      1. associate-*r/ 53.4%

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

      2. associate-/r/ 42.5%

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

   9. Simplified 42.5%

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

       1. associate-*l/ 60.6%

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

       2. associate-/r* 60.7%

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

   11. Applied egg-rr 60.7%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf 70.1%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \frac{x}{a}}{2}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;\frac{\frac{z \cdot t}{-0.2222222222222222}}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \frac{x}{a}}{2}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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) -2e+46)
   (/ (* y (/ x a)) 2.0)
   (if (<= (* x y) -0.005)
     (* -4.5 (/ (* z t) a))
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (/ (* x y) (* a 2.0))
       (* -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) {
	double tmp;
	if ((x * y) <= -2e+46) {
		tmp = (y * (x / a)) / 2.0;
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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) <= (-2d+46)) then
        tmp = (y * (x / a)) / 2.0d0
    else if ((x * y) <= (-0.005d0)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) / (a * 2.0d0)
    else
        tmp = (-4.5d0) * (t * (z / 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) <= -2e+46) {
		tmp = (y * (x / a)) / 2.0;
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / 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) <= -2e+46:
		tmp = (y * (x / a)) / 2.0
	elif (x * y) <= -0.005:
		tmp = -4.5 * ((z * t) / a)
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) / (a * 2.0)
	else:
		tmp = -4.5 * (t * (z / a))
	return tmp

Julia

x, y, z, t, a = sort([x, y, z, t, a])
function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(x * y) <= -2e+46)
		tmp = Float64(Float64(y * Float64(x / a)) / 2.0);
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / 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) <= -2e+46)
		tmp = (y * (x / a)) / 2.0;
	elseif ((x * y) <= -0.005)
		tmp = -4.5 * ((z * t) / a);
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) / (a * 2.0);
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+46], N[(N[(y * N[(x / a), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2e46

   1. Initial program 85.3%

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

   3. Taylor expanded in x around inf 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-/l* 75.7%

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

      3. associate-*r* 75.7%

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

      4. *-commutative 75.7%

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

      5. associate-*r/ 75.7%

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

   5. Simplified 75.7%

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

      1. associate-*r/ 69.5%

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

      2. associate-*l/ 77.6%

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

      3. *-commutative 77.6%

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

      4. metadata-eval 77.6%

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

      5. div-inv 77.6%

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

      6. associate-*r/ 77.6%

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

   7. Applied egg-rr 77.6%

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

   if -2e46 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 60.4%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf 70.1%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \frac{x}{a}}{2}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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) -5e+28)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -0.005)
     (* -4.5 (/ (* z t) a))
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (/ (* x y) (* a 2.0))
       (* -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) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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+28)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-0.005d0)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) / (a * 2.0d0)
    else
        tmp = (-4.5d0) * (t * (z / 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+28) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) / (a * 2.0);
	} else {
		tmp = -4.5 * (t * (z / 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+28:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -0.005:
		tmp = -4.5 * ((z * t) / a)
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) / (a * 2.0)
	else:
		tmp = -4.5 * (t * (z / 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+28)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) / Float64(a * 2.0));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / 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+28)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -0.005)
		tmp = -4.5 * ((z * t) / a);
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) / (a * 2.0);
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+28], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.99999999999999957e28

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-/l* 73.7%

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

      3. associate-*r* 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-*r/ 73.7%

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

   5. Simplified 73.7%

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

   if -4.99999999999999957e28 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 77.4%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around inf 70.1%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \end{array} \]

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) -5e+28)
   (* x (/ (* y 0.5) a))
   (if (<= (* x y) -0.005)
     (* -4.5 (/ (* z t) a))
     (if (or (<= (* x y) -5e-109) (not (<= (* x y) 1e-78)))
       (* (* x y) (/ 0.5 a))
       (* -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) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = -4.5 * (t * (z / a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
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+28)) then
        tmp = x * ((y * 0.5d0) / a)
    else if ((x * y) <= (-0.005d0)) then
        tmp = (-4.5d0) * ((z * t) / a)
    else if (((x * y) <= (-5d-109)) .or. (.not. ((x * y) <= 1d-78))) then
        tmp = (x * y) * (0.5d0 / a)
    else
        tmp = (-4.5d0) * (t * (z / a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x * y) <= -5e+28) {
		tmp = x * ((y * 0.5) / a);
	} else if ((x * y) <= -0.005) {
		tmp = -4.5 * ((z * t) / a);
	} else if (((x * y) <= -5e-109) || !((x * y) <= 1e-78)) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = -4.5 * (t * (z / 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+28:
		tmp = x * ((y * 0.5) / a)
	elif (x * y) <= -0.005:
		tmp = -4.5 * ((z * t) / a)
	elif ((x * y) <= -5e-109) or not ((x * y) <= 1e-78):
		tmp = (x * y) * (0.5 / a)
	else:
		tmp = -4.5 * (t * (z / 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+28)
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	elseif (Float64(x * y) <= -0.005)
		tmp = Float64(-4.5 * Float64(Float64(z * t) / a));
	elseif ((Float64(x * y) <= -5e-109) || !(Float64(x * y) <= 1e-78))
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	else
		tmp = Float64(-4.5 * Float64(t * Float64(z / 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+28)
		tmp = x * ((y * 0.5) / a);
	elseif ((x * y) <= -0.005)
		tmp = -4.5 * ((z * t) / a);
	elseif (((x * y) <= -5e-109) || ~(((x * y) <= 1e-78)))
		tmp = (x * y) * (0.5 / a);
	else
		tmp = -4.5 * (t * (z / a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+28], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -0.005], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-109], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e-78]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -4.99999999999999957e28

   1. Initial program 86.5%

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

   3. Taylor expanded in x around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-/l* 73.7%

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

      3. associate-*r* 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-*r/ 73.7%

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

   5. Simplified 73.7%

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

   if -4.99999999999999957e28 < (*.f64 x y) < -0.0050000000000000001

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 77.4%

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

   if -0.0050000000000000001 < (*.f64 x y) < -5.0000000000000002e-109 or 9.99999999999999999e-79 < (*.f64 x y)

   1. Initial program 96.9%

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

      1. div-inv 96.7%

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

      2. fma-neg 96.7%

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

      3. *-commutative 96.7%

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

      4. distribute-rgt-neg-in 96.7%

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

      5. distribute-rgt-neg-in 96.7%

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

      6. metadata-eval 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-/r* 96.7%

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

      9. metadata-eval 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around inf 70.0%

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

   if -5.0000000000000002e-109 < (*.f64 x y) < 9.99999999999999999e-79

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0 84.2%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+28}:\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{elif}\;x \cdot y \leq -0.005:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-109} \lor \neg \left(x \cdot y \leq 10^{-78}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+271}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - t\_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{a \cdot -0.2222222222222222}\\ \end{array} \end{array} \]

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 -6e+271)
     (* -4.5 (* t (/ z a)))
     (if (<= t_1 5e+285)
       (/ (- (* x y) t_1) (* a 2.0))
       (* t (/ z (* a -0.2222222222222222)))))))

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 <= -6e+271) {
		tmp = -4.5 * (t * (z / a));
	} else if (t_1 <= 5e+285) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = t * (z / (a * -0.2222222222222222));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-6d+271)) then
        tmp = (-4.5d0) * (t * (z / a))
    else if (t_1 <= 5d+285) then
        tmp = ((x * y) - t_1) / (a * 2.0d0)
    else
        tmp = t * (z / (a * (-0.2222222222222222d0)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t && t < a;
public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -6e+271) {
		tmp = -4.5 * (t * (z / a));
	} else if (t_1 <= 5e+285) {
		tmp = ((x * y) - t_1) / (a * 2.0);
	} else {
		tmp = t * (z / (a * -0.2222222222222222));
	}
	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 <= -6e+271:
		tmp = -4.5 * (t * (z / a))
	elif t_1 <= 5e+285:
		tmp = ((x * y) - t_1) / (a * 2.0)
	else:
		tmp = t * (z / (a * -0.2222222222222222))
	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 <= -6e+271)
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	elseif (t_1 <= 5e+285)
		tmp = Float64(Float64(Float64(x * y) - t_1) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(z / Float64(a * -0.2222222222222222)));
	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 <= -6e+271)
		tmp = -4.5 * (t * (z / a));
	elseif (t_1 <= 5e+285)
		tmp = ((x * y) - t_1) / (a * 2.0);
	else
		tmp = t * (z / (a * -0.2222222222222222));
	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, -6e+271], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+285], N[(N[(N[(x * y), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(z / N[(a * -0.2222222222222222), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -6.0000000000000001e271

   1. Initial program 58.1%

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

   3. Taylor expanded in x around 0 58.1%

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

      1. associate-/l* 93.6%

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

   5. Simplified 93.6%

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

   if -6.0000000000000001e271 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.00000000000000016e285

   1. Initial program 96.7%

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

   if 5.00000000000000016e285 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. associate-*r/ 67.3%

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

      2. associate-*r* 67.3%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r/ 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. times-frac 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/r/ 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -6 \cdot 10^{+271}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z}{a \cdot -0.2222222222222222}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+28} \lor \neg \left(x \leq 5 \cdot 10^{-128}\right):\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \end{array} \]

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
 (if (or (<= x -4e+28) (not (<= x 5e-128)))
   (* x (/ (* y 0.5) a))
   (* -4.5 (/ (* 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 tmp;
	if ((x <= -4e+28) || !(x <= 5e-128)) {
		tmp = x * ((y * 0.5) / a);
	} else {
		tmp = -4.5 * ((z * t) / 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 <= (-4d+28)) .or. (.not. (x <= 5d-128))) then
        tmp = x * ((y * 0.5d0) / a)
    else
        tmp = (-4.5d0) * ((z * t) / 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 <= -4e+28) || !(x <= 5e-128)) {
		tmp = x * ((y * 0.5) / a);
	} else {
		tmp = -4.5 * ((z * t) / 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 <= -4e+28) or not (x <= 5e-128):
		tmp = x * ((y * 0.5) / a)
	else:
		tmp = -4.5 * ((z * t) / 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 ((x <= -4e+28) || !(x <= 5e-128))
		tmp = Float64(x * Float64(Float64(y * 0.5) / a));
	else
		tmp = Float64(-4.5 * Float64(Float64(z * 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)
	tmp = 0.0;
	if ((x <= -4e+28) || ~((x <= 5e-128)))
		tmp = x * ((y * 0.5) / a);
	else
		tmp = -4.5 * ((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_] := If[Or[LessEqual[x, -4e+28], N[Not[LessEqual[x, 5e-128]], $MachinePrecision]], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999983e28 or 5.0000000000000001e-128 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf 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* 63.0%

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

      3. associate-*r* 63.0%

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

      4. *-commutative 63.0%

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

      5. associate-*r/ 63.0%

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

   5. Simplified 63.0%

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

   if -3.99999999999999983e28 < x < 5.0000000000000001e-128

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0 72.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+28} \lor \neg \left(x \leq 5 \cdot 10^{-128}\right):\\ \;\;\;\;x \cdot \frac{y \cdot 0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t, a] = \mathsf{sort}([x, y, z, t, a])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+28} \lor \neg \left(x \leq 1.4 \cdot 10^{-126}\right):\\ \;\;\;\;x \cdot \frac{0.5}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \end{array} \]

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
 (if (or (<= x -3.1e+28) (not (<= x 1.4e-126)))
   (* x (/ 0.5 (/ a y)))
   (* -4.5 (/ (* 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 tmp;
	if ((x <= -3.1e+28) || !(x <= 1.4e-126)) {
		tmp = x * (0.5 / (a / y));
	} else {
		tmp = -4.5 * ((z * t) / 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 <= (-3.1d+28)) .or. (.not. (x <= 1.4d-126))) then
        tmp = x * (0.5d0 / (a / y))
    else
        tmp = (-4.5d0) * ((z * t) / 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 <= -3.1e+28) || !(x <= 1.4e-126)) {
		tmp = x * (0.5 / (a / y));
	} else {
		tmp = -4.5 * ((z * t) / 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 <= -3.1e+28) or not (x <= 1.4e-126):
		tmp = x * (0.5 / (a / y))
	else:
		tmp = -4.5 * ((z * t) / 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 ((x <= -3.1e+28) || !(x <= 1.4e-126))
		tmp = Float64(x * Float64(0.5 / Float64(a / y)));
	else
		tmp = Float64(-4.5 * Float64(Float64(z * 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)
	tmp = 0.0;
	if ((x <= -3.1e+28) || ~((x <= 1.4e-126)))
		tmp = x * (0.5 / (a / y));
	else
		tmp = -4.5 * ((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_] := If[Or[LessEqual[x, -3.1e+28], N[Not[LessEqual[x, 1.4e-126]], $MachinePrecision]], N[(x * N[(0.5 / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1000000000000001e28 or 1.39999999999999996e-126 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf 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* 63.0%

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

      3. associate-*r* 63.0%

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

      4. *-commutative 63.0%

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

      5. associate-*r/ 63.0%

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

   5. Simplified 63.0%

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

      1. clear-num 62.3%

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

      2. un-div-inv 62.4%

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

      3. *-un-lft-identity 62.4%

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

      4. times-frac 62.4%

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

      5. metadata-eval 62.4%

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

   7. Applied egg-rr 62.4%

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

      1. associate-/r* 62.4%

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

   9. Simplified 62.4%

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

       1. div-inv 62.4%

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

       2. metadata-eval 62.4%

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

       3. associate-/l* 62.3%

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

   11. Applied egg-rr 62.3%

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

   if -3.1000000000000001e28 < x < 1.39999999999999996e-126

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0 72.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+28} \lor \neg \left(x \leq 1.4 \cdot 10^{-126}\right):\\ \;\;\;\;x \cdot \frac{0.5}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.4% 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 92.7%

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

3. Taylor expanded in x around 0 50.7%

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

   1. associate-/l* 51.3%

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

5. Simplified 51.3%

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

Developer target: 93.1% 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 2024091 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (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)))))

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