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

Average Error: 7.5 → 4.1

Time: 17.3s

Precision: binary64

Cost: 2120

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \end{array} \]

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 (+ (* x y) (* t (* z -9.0)))))
   (if (<= t_1 (- INFINITY))
     (* z (* t (/ -4.5 a)))
     (if (<= t_1 5e+302) (/ t_1 (* a 2.0)) (* -4.5 (/ z (/ a 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 = (x * y) + (t * (z * -9.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * (t * (-4.5 / a));
	} else if (t_1 <= 5e+302) {
		tmp = t_1 / (a * 2.0);
	} else {
		tmp = -4.5 * (z / (a / t));
	}
	return tmp;
}

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 = (x * y) + (t * (z * -9.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = z * (t * (-4.5 / a));
	} else if (t_1 <= 5e+302) {
		tmp = t_1 / (a * 2.0);
	} else {
		tmp = -4.5 * (z / (a / 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 = (x * y) + (t * (z * -9.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = z * (t * (-4.5 / a))
	elif t_1 <= 5e+302:
		tmp = t_1 / (a * 2.0)
	else:
		tmp = -4.5 * (z / (a / 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(x * y) + Float64(t * Float64(z * -9.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(t * Float64(-4.5 / a)));
	elseif (t_1 <= 5e+302)
		tmp = Float64(t_1 / Float64(a * 2.0));
	else
		tmp = Float64(-4.5 * Float64(z / Float64(a / 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 = (x * y) + (t * (z * -9.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = z * (t * (-4.5 / a));
	elseif (t_1 <= 5e+302)
		tmp = t_1 / (a * 2.0);
	else
		tmp = -4.5 * (z / (a / 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[(x * y), $MachinePrecision] + N[(t * N[(z * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(t * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(t$95$1 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(z / N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	7.5
Target	5.5
Herbie	4.1

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 63.7

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

      [Start] 64.0	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      sub-neg [=>] 64.0	\[ \frac{\color{blue}{x \cdot y + \left(-\left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      remove-double-neg [<=] 64.0	\[ \frac{\color{blue}{\left(-\left(-x \cdot y\right)\right)} + \left(-\left(z \cdot 9\right) \cdot t\right)}{a \cdot 2} \]
      distribute-neg-in [<=] 64.0	\[ \frac{\color{blue}{-\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)}}{a \cdot 2} \]
      +-commutative [<=] 64.0	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)}}{a \cdot 2} \]
      sub-neg [<=] 64.0	\[ \frac{-\color{blue}{\left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      neg-mul-1 [=>] 64.0	\[ \frac{\color{blue}{-1 \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)}}{a \cdot 2} \]
      associate-/l* [=>] 64.0	\[ \color{blue}{\frac{-1}{\frac{a \cdot 2}{\left(z \cdot 9\right) \cdot t - x \cdot y}}} \]
      associate-/r/ [=>] 64.0	\[ \color{blue}{\frac{-1}{a \cdot 2} \cdot \left(\left(z \cdot 9\right) \cdot t - x \cdot y\right)} \]
      sub-neg [=>] 64.0	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(z \cdot 9\right) \cdot t + \left(-x \cdot y\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(\left(-x \cdot y\right) + \left(z \cdot 9\right) \cdot t\right)} \]
      neg-sub0 [=>] 64.0	\[ \frac{-1}{a \cdot 2} \cdot \left(\color{blue}{\left(0 - x \cdot y\right)} + \left(z \cdot 9\right) \cdot t\right) \]
      associate-+l- [=>] 64.0	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(0 - \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      sub0-neg [=>] 64.0	\[ \frac{-1}{a \cdot 2} \cdot \color{blue}{\left(-\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)\right)} \]
      distribute-rgt-neg-out [=>] 64.0	\[ \color{blue}{-\frac{-1}{a \cdot 2} \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]
      distribute-lft-neg-in [=>] 64.0	\[ \color{blue}{\left(-\frac{-1}{a \cdot 2}\right) \cdot \left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \]

   3. Taylor expanded in x around 0 63.0

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

   4. Simplified 30.0

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

      [Start] 63.0	\[ -4.5 \cdot \frac{t \cdot z}{a} \]
      associate-/l* [=>] 29.9	\[ -4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}} \]
      associate-/r/ [=>] 30.0	\[ -4.5 \cdot \color{blue}{\left(\frac{t}{a} \cdot z\right)} \]

   5. Taylor expanded in t around 0 63.0

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

   6. Simplified 29.9

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

      [Start] 63.0	\[ -4.5 \cdot \frac{t \cdot z}{a} \]
      *-commutative [=>] 63.0	\[ \color{blue}{\frac{t \cdot z}{a} \cdot -4.5} \]
      associate-/r/ [<=] 63.0	\[ \color{blue}{\frac{t \cdot z}{\frac{a}{-4.5}}} \]
      associate-/l* [=>] 29.9	\[ \color{blue}{\frac{t}{\frac{\frac{a}{-4.5}}{z}}} \]

   7. Applied egg-rr 30.0

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 5e302

   1. Initial program 0.8

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

   if 5e302 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 60.7

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

   2. Simplified 60.3

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

      [Start] 60.7	\[ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
      associate-*l* [=>] 60.3	\[ \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2} \]

   3. Taylor expanded in x around 0 61.8

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

   4. Simplified 32.2

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

      [Start] 61.8	\[ -4.5 \cdot \frac{t \cdot z}{a} \]
      *-commutative [=>] 61.8	\[ -4.5 \cdot \frac{\color{blue}{z \cdot t}}{a} \]
      associate-/l* [=>] 32.2	\[ -4.5 \cdot \color{blue}{\frac{z}{\frac{a}{t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 4.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -\infty:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{x \cdot y + t \cdot \left(z \cdot -9\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.8
Cost	1612

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-200}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + -9 \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 2
Error	6.8
Cost	1612

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t \cdot -9\right)}{a \cdot 2}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+203}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Error	27.6
Cost	1372

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \leq -1.52 \cdot 10^{+58}:\\ \;\;\;\;z \cdot \left(-4.5 \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{+22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-42}:\\ \;\;\;\;z \cdot \frac{t \cdot -4.5}{a}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-135}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \end{array} \]

Alternative 4
Error	26.1
Cost	1243

\[\begin{array}{l} \mathbf{if}\;z \leq -24000000000 \lor \neg \left(z \leq -4.1 \cdot 10^{-20} \lor \neg \left(z \leq -1.38 \cdot 10^{-55}\right) \land \left(z \leq -9.8 \cdot 10^{-217} \lor \neg \left(z \leq -4.2 \cdot 10^{-279}\right) \land z \leq 1.12 \cdot 10^{-79}\right)\right):\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 5
Error	26.1
Cost	1241

\[\begin{array}{l} t_1 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ t_2 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -28000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-217} \lor \neg \left(z \leq -4.2 \cdot 10^{-279}\right) \land z \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \end{array} \]

Alternative 6
Error	26.1
Cost	1241

\[\begin{array}{l} t_1 := y \cdot \frac{x \cdot 0.5}{a}\\ t_2 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -26000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-279} \lor \neg \left(z \leq 6.6 \cdot 10^{-78}\right):\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	26.2
Cost	1241

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ t_2 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-279} \lor \neg \left(z \leq 3.6 \cdot 10^{-77}\right):\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	26.2
Cost	1240

\[\begin{array}{l} t_1 := \left(x \cdot y\right) \cdot \frac{0.5}{a}\\ t_2 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-217}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-279}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{z \cdot -4.5}{a}\\ \end{array} \]

Alternative 9
Error	7.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+252}:\\ \;\;\;\;\frac{t \cdot -4.5}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+76}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + -9 \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 10
Error	25.3
Cost	976

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ t_2 := -4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	24.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-141}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+228}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \end{array} \]

Alternative 12
Error	24.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-141}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+228}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \end{array} \]

Alternative 13
Error	32.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-85}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{+265}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \end{array} \]

Alternative 14
Error	32.9
Cost	448

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

Alternative 15
Error	32.9
Cost	448

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

Reproduce

herbie shell --seed 2023027 
(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)))