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

Average Error: 20.7 → 7.7

Time: 23.4s

Precision: binary64

Cost: 11212

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ t_2 := a \cdot \left(t \cdot -4\right)\\ t_3 := \frac{t_2 + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{t_2 + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (/ (+ (+ (* (* x 9.0) y) (* a (* t (* z -4.0)))) b) (* z c)))
        (t_2 (* a (* t -4.0)))
        (t_3 (/ (+ t_2 (/ (+ b (* x (* 9.0 y))) z)) c)))
   (if (<= t_1 (- INFINITY))
     t_3
     (if (<= t_1 -4e+14)
       t_1
       (if (<= t_1 5e+52)
         (/ (+ t_2 (/ (fma x (* 9.0 y) b) z)) c)
         (if (<= t_1 1e+305) t_1 t_3))))))

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 = ((((x * 9.0) * y) + (a * (t * (z * -4.0)))) + b) / (z * c);
	double t_2 = a * (t * -4.0);
	double t_3 = (t_2 + ((b + (x * (9.0 * y))) / z)) / c;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_1 <= -4e+14) {
		tmp = t_1;
	} else if (t_1 <= 5e+52) {
		tmp = (t_2 + (fma(x, (9.0 * y), b) / z)) / c;
	} else if (t_1 <= 1e+305) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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(Float64(Float64(Float64(Float64(x * 9.0) * y) + Float64(a * Float64(t * Float64(z * -4.0)))) + b) / Float64(z * c))
	t_2 = Float64(a * Float64(t * -4.0))
	t_3 = Float64(Float64(t_2 + Float64(Float64(b + Float64(x * Float64(9.0 * y))) / z)) / c)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_1 <= -4e+14)
		tmp = t_1;
	elseif (t_1 <= 5e+52)
		tmp = Float64(Float64(t_2 + Float64(fma(x, Float64(9.0 * y), b) / z)) / c);
	elseif (t_1 <= 1e+305)
		tmp = t_1;
	else
		tmp = t_3;
	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[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] + N[(a * N[(t * N[(z * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 + N[(N[(b + N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$3, If[LessEqual[t$95$1, -4e+14], t$95$1, If[LessEqual[t$95$1, 5e+52], N[(N[(t$95$2 + N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], t$95$1, t$95$3]]]]]]]

Target

Original	20.7
Target	14.6
Herbie	7.7

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -inf.0 or 9.9999999999999994e304 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 63.6

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

   2. Simplified 27.8

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

      [Start] 63.6	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 59.9	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]
      +-commutative [=>] 59.9	\[ \frac{\frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z}}{c} \]
      associate-+r- [=>] 59.9	\[ \frac{\frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z}}{c} \]
      div-sub [=>] 59.9	\[ \frac{\color{blue}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}}}{c} \]
      sub-neg [=>] 59.9	\[ \frac{\color{blue}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} + \left(-\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}\right)}}{c} \]
      +-commutative [=>] 59.9	\[ \frac{\color{blue}{\left(-\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}}{c} \]
      *-commutative [=>] 59.9	\[ \frac{\left(-\frac{\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*l* [=>] 59.7	\[ \frac{\left(-\frac{a \cdot \color{blue}{\left(z \cdot \left(4 \cdot t\right)\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*r* [=>] 47.4	\[ \frac{\left(-\frac{\color{blue}{\left(a \cdot z\right) \cdot \left(4 \cdot t\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*l/ [<=] 38.3	\[ \frac{\left(-\color{blue}{\frac{a \cdot z}{z} \cdot \left(4 \cdot t\right)}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      distribute-rgt-neg-in [=>] 38.3	\[ \frac{\color{blue}{\frac{a \cdot z}{z} \cdot \left(-4 \cdot t\right)} + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-/l* [=>] 27.8	\[ \frac{\color{blue}{\frac{a}{\frac{z}{z}}} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      *-inverses [=>] 27.8	\[ \frac{\frac{a}{\color{blue}{1}} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      /-rgt-identity [=>] 27.8	\[ \frac{\color{blue}{a} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      *-commutative [=>] 27.8	\[ \frac{a \cdot \left(-\color{blue}{t \cdot 4}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      distribute-rgt-neg-in [=>] 27.8	\[ \frac{a \cdot \color{blue}{\left(t \cdot \left(-4\right)\right)} + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      metadata-eval [=>] 27.8	\[ \frac{a \cdot \left(t \cdot \color{blue}{-4}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      +-commutative [=>] 27.8	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + b}}{z}}{c} \]
      associate-*l* [=>] 27.8	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} + b}{z}}{c} \]
      fma-def [=>] 27.8	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}}{z}}{c} \]

   3. Applied egg-rr 27.8

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

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -4e14 or 5e52 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 9.9999999999999994e304

   1. Initial program 0.6

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

   if -4e14 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 5e52

   1. Initial program 12.3

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

   2. Simplified 1.2

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

      [Start] 12.3	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 1.4	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]
      +-commutative [=>] 1.4	\[ \frac{\frac{\color{blue}{b + \left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)}}{z}}{c} \]
      associate-+r- [=>] 1.4	\[ \frac{\frac{\color{blue}{\left(b + \left(x \cdot 9\right) \cdot y\right) - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a}}{z}}{c} \]
      div-sub [=>] 1.4	\[ \frac{\color{blue}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} - \frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}}}{c} \]
      sub-neg [=>] 1.4	\[ \frac{\color{blue}{\frac{b + \left(x \cdot 9\right) \cdot y}{z} + \left(-\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}\right)}}{c} \]
      +-commutative [=>] 1.4	\[ \frac{\color{blue}{\left(-\frac{\left(\left(z \cdot 4\right) \cdot t\right) \cdot a}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}}{c} \]
      *-commutative [=>] 1.4	\[ \frac{\left(-\frac{\color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*l* [=>] 1.4	\[ \frac{\left(-\frac{a \cdot \color{blue}{\left(z \cdot \left(4 \cdot t\right)\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*r* [=>] 6.3	\[ \frac{\left(-\frac{\color{blue}{\left(a \cdot z\right) \cdot \left(4 \cdot t\right)}}{z}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-*l/ [<=] 6.3	\[ \frac{\left(-\color{blue}{\frac{a \cdot z}{z} \cdot \left(4 \cdot t\right)}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      distribute-rgt-neg-in [=>] 6.3	\[ \frac{\color{blue}{\frac{a \cdot z}{z} \cdot \left(-4 \cdot t\right)} + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      associate-/l* [=>] 1.2	\[ \frac{\color{blue}{\frac{a}{\frac{z}{z}}} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      *-inverses [=>] 1.2	\[ \frac{\frac{a}{\color{blue}{1}} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      /-rgt-identity [=>] 1.2	\[ \frac{\color{blue}{a} \cdot \left(-4 \cdot t\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      *-commutative [=>] 1.2	\[ \frac{a \cdot \left(-\color{blue}{t \cdot 4}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      distribute-rgt-neg-in [=>] 1.2	\[ \frac{a \cdot \color{blue}{\left(t \cdot \left(-4\right)\right)} + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      metadata-eval [=>] 1.2	\[ \frac{a \cdot \left(t \cdot \color{blue}{-4}\right) + \frac{b + \left(x \cdot 9\right) \cdot y}{z}}{c} \]
      +-commutative [=>] 1.2	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{\left(x \cdot 9\right) \cdot y + b}}{z}}{c} \]
      associate-*l* [=>] 1.2	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} + b}{z}}{c} \]
      fma-def [=>] 1.2	\[ \frac{a \cdot \left(t \cdot -4\right) + \frac{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}}{z}}{c} \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 10^{+305}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	6354

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq -4 \cdot 10^{-21} \lor \neg \left(t_1 \leq 5 \cdot 10^{+52}\right) \land t_1 \leq 10^{+305}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	14.1
Cost	1616

\[\begin{array}{l} t_1 := 9 \cdot \left(x \cdot y\right)\\ t_2 := a \cdot \left(t \cdot -4\right)\\ t_3 := \frac{t_2 + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-131}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{b + t_1}{z \cdot c}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{t_2 + \frac{t_1}{z}}{c}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{-4 \cdot \frac{a \cdot \left(z \cdot t\right)}{c} + \frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	25.9
Cost	1492

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{9}{\frac{c}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{b + 9 \cdot \left(x \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+271}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{y}{\frac{z}{x}}}{c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z}{\frac{x}{c}}}\\ \end{array} \]

Alternative 4
Error	14.6
Cost	1484

\[\begin{array}{l} t_1 := a \cdot \left(z \cdot t\right)\\ t_2 := \frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{9 \cdot \left(x \cdot y\right) + -4 \cdot t_1}{z \cdot c}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{-4 \cdot \frac{t_1}{c} + \frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	9.7
Cost	1481

\[\begin{array}{l} t_1 := x \cdot \left(9 \cdot y\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{-17} \lor \neg \left(z \leq 9.5 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + t_1}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(t_1 - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right)}{z \cdot c}\\ \end{array} \]

Alternative 6
Error	35.6
Cost	1372

\[\begin{array}{l} t_1 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-157}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;a \leq 6.9 \cdot 10^{-171}:\\ \;\;\;\;\frac{1}{c \cdot \frac{z}{b}}\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;a \leq 4000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 7
Error	21.9
Cost	1364

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-234}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{-4 \cdot \frac{a \cdot \left(z \cdot t\right)}{c} + \frac{b}{c}}{z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{y}{\frac{z}{x}}}{c}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+85}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	19.7
Cost	1233

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-119}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-78} \lor \neg \left(z \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c}\\ \end{array} \]

Alternative 9
Error	35.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;b \cdot \frac{\frac{1}{z}}{c}\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-241}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-238}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z}{\frac{x}{c}}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 10
Error	35.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+20}:\\ \;\;\;\;b \cdot \frac{\frac{1}{c}}{z}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-241}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-238}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z}{\frac{x}{c}}}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 11
Error	27.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{-73}:\\ \;\;\;\;t \cdot \frac{a \cdot -4}{c}\\ \mathbf{elif}\;a \leq 102000000000:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+76}:\\ \;\;\;\;\frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 12
Error	34.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+22} \lor \neg \left(b \leq 9.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 13
Error	34.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+19}:\\ \;\;\;\;b \cdot \frac{\frac{1}{z}}{c}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-17}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 14
Error	43.5
Cost	320

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

Reproduce

herbie shell --seed 2022364 
(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)))