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

Average Error: 8.1 → 0.8

Time: 18.6s

Precision: binary64

Cost: 3664

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ [z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot 0.5}{\frac{a}{x}}\\ t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ t_3 := \frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+282}:\\ \;\;\;\;t_1 + t_3\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-241}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + t_3\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y 0.5) (/ a x)))
        (t_2 (+ (* x y) (* t (* z -9.0))))
        (t_3 (/ (* t -4.5) (/ a z))))
   (if (<= t_2 -5e+282)
     (+ t_1 t_3)
     (if (<= t_2 -2e-241)
       (+ (* -4.5 (/ (* z t) a)) (* 0.5 (/ (* x y) a)))
       (if (<= t_2 5e-103)
         (+ (* x (* y (/ 0.5 a))) t_3)
         (if (<= t_2 5e+282)
           (/ (+ (* x y) (* z (* t -9.0))) (* a 2.0))
           (+ t_1 (* (/ z a) (/ -9.0 (/ 2.0 t))))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * 0.5) / (a / x);
	double t_2 = (x * y) + (t * (z * -9.0));
	double t_3 = (t * -4.5) / (a / z);
	double tmp;
	if (t_2 <= -5e+282) {
		tmp = t_1 + t_3;
	} else if (t_2 <= -2e-241) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else if (t_2 <= 5e-103) {
		tmp = (x * (y * (0.5 / a))) + t_3;
	} else if (t_2 <= 5e+282) {
		tmp = ((x * y) + (z * (t * -9.0))) / (a * 2.0);
	} else {
		tmp = t_1 + ((z / a) * (-9.0 / (2.0 / t)));
	}
	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
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * 0.5d0) / (a / x)
    t_2 = (x * y) + (t * (z * (-9.0d0)))
    t_3 = (t * (-4.5d0)) / (a / z)
    if (t_2 <= (-5d+282)) then
        tmp = t_1 + t_3
    else if (t_2 <= (-2d-241)) then
        tmp = ((-4.5d0) * ((z * t) / a)) + (0.5d0 * ((x * y) / a))
    else if (t_2 <= 5d-103) then
        tmp = (x * (y * (0.5d0 / a))) + t_3
    else if (t_2 <= 5d+282) then
        tmp = ((x * y) + (z * (t * (-9.0d0)))) / (a * 2.0d0)
    else
        tmp = t_1 + ((z / a) * ((-9.0d0) / (2.0d0 / t)))
    end if
    code = tmp
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);
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * 0.5) / (a / x);
	double t_2 = (x * y) + (t * (z * -9.0));
	double t_3 = (t * -4.5) / (a / z);
	double tmp;
	if (t_2 <= -5e+282) {
		tmp = t_1 + t_3;
	} else if (t_2 <= -2e-241) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else if (t_2 <= 5e-103) {
		tmp = (x * (y * (0.5 / a))) + t_3;
	} else if (t_2 <= 5e+282) {
		tmp = ((x * y) + (z * (t * -9.0))) / (a * 2.0);
	} else {
		tmp = t_1 + ((z / a) * (-9.0 / (2.0 / t)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = (y * 0.5) / (a / x)
	t_2 = (x * y) + (t * (z * -9.0))
	t_3 = (t * -4.5) / (a / z)
	tmp = 0
	if t_2 <= -5e+282:
		tmp = t_1 + t_3
	elif t_2 <= -2e-241:
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a))
	elif t_2 <= 5e-103:
		tmp = (x * (y * (0.5 / a))) + t_3
	elif t_2 <= 5e+282:
		tmp = ((x * y) + (z * (t * -9.0))) / (a * 2.0)
	else:
		tmp = t_1 + ((z / a) * (-9.0 / (2.0 / t)))
	return tmp

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * 0.5) / Float64(a / x))
	t_2 = Float64(Float64(x * y) + Float64(t * Float64(z * -9.0)))
	t_3 = Float64(Float64(t * -4.5) / Float64(a / z))
	tmp = 0.0
	if (t_2 <= -5e+282)
		tmp = Float64(t_1 + t_3);
	elseif (t_2 <= -2e-241)
		tmp = Float64(Float64(-4.5 * Float64(Float64(z * t) / a)) + Float64(0.5 * Float64(Float64(x * y) / a)));
	elseif (t_2 <= 5e-103)
		tmp = Float64(Float64(x * Float64(y * Float64(0.5 / a))) + t_3);
	elseif (t_2 <= 5e+282)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t * -9.0))) / Float64(a * 2.0));
	else
		tmp = Float64(t_1 + Float64(Float64(z / a) * Float64(-9.0 / Float64(2.0 / t))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * 0.5) / (a / x);
	t_2 = (x * y) + (t * (z * -9.0));
	t_3 = (t * -4.5) / (a / z);
	tmp = 0.0;
	if (t_2 <= -5e+282)
		tmp = t_1 + t_3;
	elseif (t_2 <= -2e-241)
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	elseif (t_2 <= 5e-103)
		tmp = (x * (y * (0.5 / a))) + t_3;
	elseif (t_2 <= 5e+282)
		tmp = ((x * y) + (z * (t * -9.0))) / (a * 2.0);
	else
		tmp = t_1 + ((z / a) * (-9.0 / (2.0 / t)));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * 0.5), $MachinePrecision] / N[(a / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * -4.5), $MachinePrecision] / N[(a / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+282], N[(t$95$1 + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, -2e-241], N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-103], N[(N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], If[LessEqual[t$95$2, 5e+282], N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(z / a), $MachinePrecision] * N[(-9.0 / N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Target

Original	8.1
Target	5.8
Herbie	0.8

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

Derivation

1. Split input into 5 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -4.99999999999999978e282

   1. Initial program 52.1

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

   2. Simplified 51.4

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

      [Start] 52.1	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 51.4	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 26.6

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

   4. Simplified 0.9

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

      [Start] 26.6	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z}{a} \cdot \frac{9 \cdot t}{2}\right) \]
      sub-neg [<=] 26.6	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
      associate-*l* [=>] 0.9	\[ \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} - \frac{z}{a} \cdot \frac{9 \cdot t}{2} \]
      associate-/l* [=>] 0.9	\[ x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{a} \cdot \color{blue}{\frac{9}{\frac{2}{t}}} \]

   5. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{y \cdot 0.5}{\frac{a}{x}}} - \frac{z}{a} \cdot \frac{9}{\frac{2}{t}} \]

   6. Applied egg-rr 0.9

      \[\leadsto \frac{y \cdot 0.5}{\frac{a}{x}} - \color{blue}{\frac{t \cdot 4.5}{\frac{a}{z}}} \]

   if -4.99999999999999978e282 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -1.9999999999999999e-241

   1. Initial program 0.3

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

   2. Simplified 0.4

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

      [Start] 0.3	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 0.3	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      remove-double-neg [<=] 0.3	\[ \frac{\color{blue}{\left(-\left(-x \cdot y\right)\right)} + \left(-\left(z \cdot 9\right) \cdot t\right)}{a \cdot 2} \]
      distribute-neg-in [<=] 0.3	\[ \frac{\color{blue}{-\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [<=] 0.3	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)}}{a \cdot 2} \]
      sub-neg [<=] 0.3	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 0.3	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 0.6	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 0.4	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      sub-neg [=>] 0.4	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \]
      +-commutative [=>] 0.4	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \]
      neg-sub0 [=>] 0.4	\[ \frac{-1}{a \cdot 2} \cdot \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \]
      associate-+l- [=>] 0.4	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      sub0-neg [=>] 0.4	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      distribute-rgt-neg-out [=>] 0.4	\[ \color{blue}{-\frac{-1}{a \cdot 2} \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]
      distribute-lft-neg-in [=>] 0.4	\[ \color{blue}{\left(-\frac{-1}{a \cdot 2}\right) \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

   3. Taylor expanded in x around 0 0.3

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

   if -1.9999999999999999e-241 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.99999999999999966e-103

   1. Initial program 5.2

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

   2. Simplified 5.2

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

      [Start] 5.2	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 5.2	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 4.3

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

   4. Simplified 4.2

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

      [Start] 4.3	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z}{a} \cdot \frac{9 \cdot t}{2}\right) \]
      sub-neg [<=] 4.3	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
      associate-*l* [=>] 4.2	\[ \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} - \frac{z}{a} \cdot \frac{9 \cdot t}{2} \]
      associate-/l* [=>] 4.2	\[ x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{a} \cdot \color{blue}{\frac{9}{\frac{2}{t}}} \]

   5. Applied egg-rr 3.9

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

   if 4.99999999999999966e-103 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.99999999999999978e282

   1. Initial program 0.3

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

   2. Simplified 0.4

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

      [Start] 0.3	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 0.4	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   if 4.99999999999999978e282 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 54.1

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

   2. Simplified 54.0

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

      [Start] 54.1	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 54.0	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Applied egg-rr 30.9

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

   4. Simplified 0.4

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

      [Start] 30.9	\[ \left(x \cdot y\right) \cdot \frac{0.5}{a} + \left(-\frac{z}{a} \cdot \frac{9 \cdot t}{2}\right) \]
      sub-neg [<=] 30.9	\[ \color{blue}{\left(x \cdot y\right) \cdot \frac{0.5}{a} - \frac{z}{a} \cdot \frac{9 \cdot t}{2}} \]
      associate-*l* [=>] 0.5	\[ \color{blue}{x \cdot \left(y \cdot \frac{0.5}{a}\right)} - \frac{z}{a} \cdot \frac{9 \cdot t}{2} \]
      associate-/l* [=>] 0.4	\[ x \cdot \left(y \cdot \frac{0.5}{a}\right) - \frac{z}{a} \cdot \color{blue}{\frac{9}{\frac{2}{t}}} \]

   5. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{y \cdot 0.5}{\frac{a}{x}}} - \frac{z}{a} \cdot \frac{9}{\frac{2}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -5 \cdot 10^{+282}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}} + \frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -2 \cdot 10^{-241}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 5 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{\frac{a}{x}} + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	3537

\[\begin{array}{l} t_1 := x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{t \cdot -4.5}{\frac{a}{z}}\\ t_2 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-241}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-103} \lor \neg \left(t_2 \leq 5 \cdot 10^{+282}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	3536

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ t_2 := \frac{t \cdot -4.5}{\frac{a}{z}}\\ t_3 := \frac{y \cdot 0.5}{\frac{a}{x}} + t_2\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+282}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-241}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	4.4
Cost	2632

\[\begin{array}{l} t_1 := \frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{-4.5}{\frac{a}{t}}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Error	4.4
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+252}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 5
Error	24.5
Cost	1241

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-45}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-22} \lor \neg \left(y \leq 125000000000\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	24.5
Cost	1240

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-46}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 12000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	24.4
Cost	1240

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_2 := y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-45}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 520000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	24.4
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-193}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-45}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 30500000000:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	24.4
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-193}:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-172}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-45}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 42000000000:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.4
Cost	1240

\[\begin{array}{l} t_1 := y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-193}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\left(z \cdot t\right) \cdot -9\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-172}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{elif}\;y \leq 10^{-44}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 10200000000000:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	7.3
Cost	1229

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+262} \lor \neg \left(y \leq 4.6 \cdot 10^{+278}\right):\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + \left(z \cdot t\right) \cdot -9\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \end{array} \]

Alternative 12
Error	23.9
Cost	977

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;x \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-108}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-121} \lor \neg \left(x \leq 4.4 \cdot 10^{-98}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 13
Error	31.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+151}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-21}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 14
Error	32.9
Cost	448

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

Alternative 15
Error	32.9
Cost	448

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

Reproduce

herbie shell --seed 2023073 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (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)))