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

Average Error: 20.2 → 7.2

Time: 30.7s

Precision: binary64

Cost: 17484

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

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

↓

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + t_1}{c}\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, t_1\right)}{c}\\ \mathbf{elif}\;t_2 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (fma x (* 9.0 y) b) z))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_2 (- INFINITY))
     (/ (+ (* a (* t -4.0)) t_1) c)
     (if (<= t_2 -5e+74)
       t_2
       (if (<= t_2 5e-48)
         (/ (fma t (* a -4.0) t_1) c)
         (if (<= t_2 4e+292)
           t_2
           (+ (* -4.0 (* t (* a (/ 1.0 c)))) (/ b (* z c)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(x, (9.0 * y), b) / z;
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((a * (t * -4.0)) + t_1) / c;
	} else if (t_2 <= -5e+74) {
		tmp = t_2;
	} else if (t_2 <= 5e-48) {
		tmp = fma(t, (a * -4.0), t_1) / c;
	} else if (t_2 <= 4e+292) {
		tmp = t_2;
	} else {
		tmp = (-4.0 * (t * (a * (1.0 / c)))) + (b / (z * c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

↓

function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(x, Float64(9.0 * y), b) / z)
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(a * Float64(t * -4.0)) + t_1) / c);
	elseif (t_2 <= -5e+74)
		tmp = t_2;
	elseif (t_2 <= 5e-48)
		tmp = Float64(fma(t, Float64(a * -4.0), t_1) / c);
	elseif (t_2 <= 4e+292)
		tmp = t_2;
	else
		tmp = Float64(Float64(-4.0 * Float64(t * Float64(a * Float64(1.0 / c)))) + Float64(b / Float64(z * c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(a * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, -5e+74], t$95$2, If[LessEqual[t$95$2, 5e-48], N[(N[(t * N[(a * -4.0), $MachinePrecision] + t$95$1), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, 4e+292], t$95$2, N[(N[(-4.0 * N[(t * N[(a * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	20.2
Target	14.5
Herbie	7.2

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -inf.0

   1. Initial program 64.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 25.1

      \[\leadsto \color{blue}{\frac{a \cdot \left(t \cdot -4\right) + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 58.4	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -4.99999999999999963e74 or 4.9999999999999999e-48 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 4.0000000000000001e292

   1. Initial program 0.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   if -4.99999999999999963e74 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 4.9999999999999999e-48

   1. Initial program 12.5

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 1.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}} \]
      Proof

      [Start] 12.5	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 1.9	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   if 4.0000000000000001e292 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 61.2

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 26.3

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}} \]
      Proof

      [Start] 61.2	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 59.5	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   3. Taylor expanded in x around 0 27.6

      \[\leadsto \color{blue}{\frac{\frac{b}{z} + -4 \cdot \left(a \cdot t\right)}{c}} \]

   4. Taylor expanded in b around 0 30.3

      \[\leadsto \color{blue}{\frac{b}{c \cdot z} + -4 \cdot \frac{a \cdot t}{c}} \]

   5. Applied egg-rr 25.1

      \[\leadsto \frac{b}{c \cdot z} + -4 \cdot \color{blue}{\left(t \cdot \left(a \cdot \frac{1}{c}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 4 \cdot 10^{+292}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.2
Cost	11212

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_2 := \frac{a \cdot \left(t \cdot -4\right) + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \end{array} \]

Alternative 2
Error	8.1
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_2 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \end{array} \]

Alternative 3
Error	8.0
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{b + \left(x \cdot \left(9 \cdot y\right) + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{z}\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{+292}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \end{array} \]

Alternative 4
Error	34.0
Cost	3044

\[\begin{array}{l} t_1 := \frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c}\\ \mathbf{if}\;x \cdot 9 \leq -1 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \left(9 \cdot \frac{y}{z \cdot c}\right)\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-188}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq 4 \cdot 10^{-275}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;x \cdot 9 \leq 5 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 5
Error	37.6
Cost	2916

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ \mathbf{if}\;x \cdot 9 \leq -1 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \left(9 \cdot \frac{y}{z \cdot c}\right)\\ \mathbf{elif}\;x \cdot 9 \leq -4 \cdot 10^{+19}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-188}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \cdot 9 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq 5 \cdot 10^{-221}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \cdot 9 \leq 5 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot \frac{9}{c}\right)\\ \end{array} \]

Alternative 6
Error	22.3
Cost	2520

\[\begin{array}{l} t_1 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ t_2 := \frac{a \cdot \left(t \cdot -4\right) + \frac{9 \cdot \left(x \cdot y\right)}{z}}{c}\\ \mathbf{if}\;x \cdot 9 \leq -4 \cdot 10^{+156}:\\ \;\;\;\;x \cdot \left(9 \cdot \frac{y}{z \cdot c}\right)\\ \mathbf{elif}\;x \cdot 9 \leq -4 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \cdot 9 \leq -4 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot 9 \leq -5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \cdot 9 \leq 2 \cdot 10^{-153}:\\ \;\;\;\;-4 \cdot \left(t \cdot \left(a \cdot \frac{1}{c}\right)\right) + \frac{b}{z \cdot c}\\ \mathbf{elif}\;x \cdot 9 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	37.3
Cost	2028

\[\begin{array}{l} t_1 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ t_2 := \frac{b}{z \cdot c}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{+18}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \leq -1.28 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-190}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-221}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-77}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c}\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-10}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	37.3
Cost	2028

\[\begin{array}{l} t_1 := 9 \cdot \frac{\frac{y}{z}}{\frac{c}{x}}\\ t_2 := \frac{b}{z \cdot c}\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{+18}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-97}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-190}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-218}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-220}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-77}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	37.8
Cost	2028

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \left(9 \cdot \frac{y}{z \cdot c}\right)\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-191}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-72}:\\ \;\;\;\;9 \cdot \frac{x \cdot y}{z \cdot c}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-12}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{\frac{y}{z}}{\frac{c}{x}}\\ \end{array} \]

Alternative 10
Error	40.1
Cost	1769

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ t_2 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.15 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-220} \lor \neg \left(x \leq 1.9 \cdot 10^{-194}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	37.2
Cost	1764

\[\begin{array}{l} t_1 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ t_2 := \frac{b}{z \cdot c}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+18}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-95}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-190}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	14.0
Cost	1481

\[\begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+93} \lor \neg \left(z \leq 2.6 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(x \cdot \left(9 \cdot y\right) + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \end{array} \]

Alternative 13
Error	35.4
Cost	976

\[\begin{array}{l} t_1 := -4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.45 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 14
Error	35.3
Cost	976

\[\begin{array}{l} t_1 := -4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 15
Error	35.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+47}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-142}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 16
Error	35.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-141}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 17
Error	34.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-135}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-176}:\\ \;\;\;\;b \cdot \frac{\frac{1}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 18
Error	35.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -4.7 \cdot 10^{+48}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{-136}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-176}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 19
Error	35.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+47}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t \cdot -4}{c}\\ \end{array} \]

Alternative 20
Error	19.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+60} \lor \neg \left(z \leq 10^{-32}\right):\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c}\\ \end{array} \]

Alternative 21
Error	43.1
Cost	320

\[\frac{b}{z \cdot c} \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))