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

Average Error: 8.0 → 0.9

Time: 22.0s

Precision: binary64

Cost: 8520

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+245}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+222}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

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 -5e+245)
     (+ (* x (* y (/ 0.5 a))) (* (/ z a) (/ -9.0 (/ 2.0 t))))
     (if (<= t_1 4e+222)
       (+ (* -4.5 (/ (* z t) a)) (* 0.5 (/ (* x y) a)))
       (fma -4.5 (/ t (/ a z)) (* 0.5 (/ y (/ a x))))))))

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 <= -5e+245) {
		tmp = (x * (y * (0.5 / a))) + ((z / a) * (-9.0 / (2.0 / t)));
	} else if (t_1 <= 4e+222) {
		tmp = (-4.5 * ((z * t) / a)) + (0.5 * ((x * y) / a));
	} else {
		tmp = fma(-4.5, (t / (a / z)), (0.5 * (y / (a / x))));
	}
	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 <= -5e+245)
		tmp = Float64(Float64(x * Float64(y * Float64(0.5 / a))) + Float64(Float64(z / a) * Float64(-9.0 / Float64(2.0 / t))));
	elseif (t_1 <= 4e+222)
		tmp = Float64(Float64(-4.5 * Float64(Float64(z * t) / a)) + Float64(0.5 * Float64(Float64(x * y) / a)));
	else
		tmp = fma(-4.5, Float64(t / Float64(a / z)), Float64(0.5 * Float64(y / Float64(a / x))));
	end
	return 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, -5e+245], N[(N[(x * N[(y * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z / a), $MachinePrecision] * N[(-9.0 / N[(2.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+222], N[(N[(-4.5 * N[(N[(z * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y / N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	8.0
Target	5.6
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -5.00000000000000034e245

   1. Initial program 39.8

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

   2. Simplified 39.7

      \[\leadsto \color{blue}{\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}} \]
      Proof
      (/.f64 (-.f64 (*.f64 x y) (*.f64 z (*.f64 9 t))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 9) t))) (*.f64 a 2)): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 20.0

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

   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
      (-.f64 (*.f64 x (*.f64 y (/.f64 1/2 a))) (*.f64 (/.f64 z a) (/.f64 9 (/.f64 2 t)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x y) (/.f64 1/2 a))) (*.f64 (/.f64 z a) (/.f64 9 (/.f64 2 t)))): 4 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (*.f64 x y) (/.f64 1/2 a)) (*.f64 (/.f64 z a) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 9 t) 2)))): 0 points increase in error, 3 points decrease in error
      (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 (*.f64 x y) (/.f64 1/2 a)) (neg.f64 (*.f64 (/.f64 z a) (/.f64 (*.f64 9 t) 2))))): 1 points increase in error, 0 points decrease in error

   if -5.00000000000000034e245 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 4.0000000000000002e222

   1. Initial program 0.9

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

   2. Simplified 1.0

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \mathsf{fma}\left(x, y, z \cdot \left(t \cdot -9\right)\right)} \]
      Proof
      (/.f64 (-.f64 (*.f64 x y) (*.f64 z (*.f64 9 t))) (*.f64 a 2)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z 9) t))) (*.f64 a 2)): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 1.0

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

   if 4.0000000000000002e222 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 33.8

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

   2. Taylor expanded in x around 0 32.9

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

   3. Simplified 0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)} \]
      Proof
      (fma.f64 -9/2 (/.f64 t (/.f64 a z)) (*.f64 1/2 (/.f64 y (/.f64 a x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 -9/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t z) a)) (*.f64 1/2 (/.f64 y (/.f64 a x)))): 4 points increase in error, 0 points decrease in error
      (fma.f64 -9/2 (/.f64 (*.f64 t z) a) (*.f64 1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) a)))): 0 points increase in error, 3 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -9/2 (/.f64 (*.f64 t z) a)) (*.f64 1/2 (/.f64 (*.f64 y x) a)))): 3 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq -5 \cdot 10^{+245}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \mathbf{elif}\;x \cdot y + t \cdot \left(z \cdot -9\right) \leq 4 \cdot 10^{+222}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4.5, \frac{t}{\frac{a}{z}}, 0.5 \cdot \frac{y}{\frac{a}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	32.3
Cost	3828

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_2 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ t_3 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \cdot 2 \leq -5 \cdot 10^{+86}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{elif}\;a \cdot 2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 0.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \cdot 2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 2
Error	32.3
Cost	3828

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ t_2 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \cdot 2 \leq -5 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{elif}\;a \cdot 2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 0.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \cdot 2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 3
Error	32.3
Cost	3828

\[\begin{array}{l} t_1 := \frac{x \cdot 0.5}{\frac{a}{y}}\\ t_2 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \cdot 2 \leq -5 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{elif}\;a \cdot 2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 0.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \cdot 2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 4
Error	32.3
Cost	3828

\[\begin{array}{l} t_1 := \frac{-4.5}{\frac{a}{z \cdot t}}\\ t_2 := \frac{x \cdot y}{a \cdot 2}\\ \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \cdot 2 \leq -5 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{elif}\;a \cdot 2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 10^{-62}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \mathbf{elif}\;a \cdot 2 \leq 0.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \cdot 2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{+135}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 5
Error	32.3
Cost	3828

\[\begin{array}{l} t_1 := \frac{-4.5}{\frac{a}{z \cdot t}}\\ t_2 := \frac{x \cdot y}{a \cdot 2}\\ \mathbf{if}\;a \cdot 2 \leq -1 \cdot 10^{+270}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{+194}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \cdot 2 \leq -5 \cdot 10^{+70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq -2 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a \cdot \frac{2}{x}}\\ \mathbf{elif}\;a \cdot 2 \leq -1 \cdot 10^{-171}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{x}{\frac{2}{y}}}{a}\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 2 \leq 10^{-62}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \mathbf{elif}\;a \cdot 2 \leq 0.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \cdot 2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 2 \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{z \cdot -4.5}{\frac{a}{t}}\\ \mathbf{elif}\;a \cdot 2 \leq 10^{+135}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 6
Error	5.2
Cost	2384

\[\begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := -4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-282}:\\ \;\;\;\;t_2 + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-63}:\\ \;\;\;\;t_2 + 0.5 \cdot \left(x \cdot \frac{1}{\frac{a}{y}}\right)\\ \mathbf{elif}\;t_1 \leq 10^{+196}:\\ \;\;\;\;\frac{x \cdot y + \left(z \cdot t\right) \cdot -9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \end{array} \]

Alternative 7
Error	0.9
Cost	2377

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+245} \lor \neg \left(t_1 \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{a}\right) + \frac{z}{a} \cdot \frac{-9}{\frac{2}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \end{array} \]

Alternative 8
Error	4.4
Cost	2248

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a} + 0.5 \cdot \frac{x \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \end{array} \]

Alternative 9
Error	32.4
Cost	2164

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_2 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_3 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ t_4 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;a \leq -1.8 \cdot 10^{+269}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -7.1 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.7 \cdot 10^{+83}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8.4 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-268}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-37}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 3200:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{+60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 10
Error	32.3
Cost	2164

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_2 := -4.5 \cdot \frac{z \cdot t}{a}\\ t_3 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ t_4 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{+261}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{+82}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.6 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-268}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-49}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 4100:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{+59}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 11
Error	32.3
Cost	2164

\[\begin{array}{l} t_1 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_2 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ t_3 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;a \leq -1.18 \cdot 10^{+269}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \leq -6.6 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+83}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-186}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-267}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-148}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 240:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+135}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 12
Error	32.2
Cost	2164

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ t_2 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_3 := \frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ t_4 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{+266}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \leq -1.12 \cdot 10^{+194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.92 \cdot 10^{+84}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-74}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq -1.5 \cdot 10^{-172}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-142}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+134}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 13
Error	32.1
Cost	2164

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ t_2 := -4.5 \cdot \frac{z}{\frac{a}{t}}\\ t_3 := 0.5 \cdot \frac{y}{\frac{a}{x}}\\ \mathbf{if}\;a \leq -7 \cdot 10^{+261}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5}{a}\\ \mathbf{elif}\;a \leq -5 \cdot 10^{+193}:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{+84}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.65 \cdot 10^{-178}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-267}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{-4.5}{\frac{a}{z \cdot t}}\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.5:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{-4.5}}{z}}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 14
Error	4.4
Cost	2120

\[\begin{array}{l} t_1 := x \cdot y + t \cdot \left(z \cdot -9\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t}{\frac{\frac{a}{z}}{-4.5}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{x \cdot y + \left(z \cdot t\right) \cdot -9}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{a}{y}}\\ \end{array} \]

Alternative 15
Error	4.6
Cost	1352

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+221}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+215}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(x \cdot y + z \cdot \left(t \cdot -9\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}}\\ \end{array} \]

Alternative 16
Error	24.1
Cost	1241

\[\begin{array}{l} t_1 := 0.5 \cdot \left(x \cdot \frac{y}{a}\right)\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-105}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+42}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+153} \lor \neg \left(t \leq 8 \cdot 10^{+290}\right):\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \end{array} \]

Alternative 17
Error	33.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+165} \lor \neg \left(t \leq 3.5 \cdot 10^{+286}\right):\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 18
Error	33.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+185}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+284}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 19
Error	32.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+182}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+284}:\\ \;\;\;\;-4.5 \cdot \frac{z}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array} \]

Alternative 20
Error	31.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+185}:\\ \;\;\;\;-4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;z \leq 10^{-28}:\\ \;\;\;\;-4.5 \cdot \frac{z \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \end{array} \]

Alternative 21
Error	33.2
Cost	448

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

Reproduce

herbie shell --seed 2022343 
(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)))