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

Average Error: 20.8 → 9.2

Time: 23.6s

Precision: binary64

Cost: 6352

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (/ (+ (/ b z) (* t (* -4.0 a))) c))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_2 (- INFINITY))
     (* (/ (* a t) c) -4.0)
     (if (<= t_2 -1e-278)
       t_2
       (if (<= t_2 2e-247) t_1 (if (<= t_2 5e+307) t_2 t_1))))))

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 = ((b / z) + (t * (-4.0 * a))) / c;
	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) / c) * -4.0;
	} else if (t_2 <= -1e-278) {
		tmp = t_2;
	} else if (t_2 <= 2e-247) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static 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);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((b / z) + (t * (-4.0 * a))) / c;
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((a * t) / c) * -4.0;
	} else if (t_2 <= -1e-278) {
		tmp = t_2;
	} else if (t_2 <= 2e-247) {
		tmp = t_1;
	} else if (t_2 <= 5e+307) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)

↓

def code(x, y, z, t, a, b, c):
	t_1 = ((b / z) + (t * (-4.0 * a))) / c
	t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((a * t) / c) * -4.0
	elif t_2 <= -1e-278:
		tmp = t_2
	elif t_2 <= 2e-247:
		tmp = t_1
	elif t_2 <= 5e+307:
		tmp = t_2
	else:
		tmp = t_1
	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(b / z) + Float64(t * Float64(-4.0 * a))) / c)
	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 * t) / c) * -4.0);
	elseif (t_2 <= -1e-278)
		tmp = t_2;
	elseif (t_2 <= 2e-247)
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
end

↓

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = ((b / z) + (t * (-4.0 * a))) / c;
	t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((a * t) / c) * -4.0;
	elseif (t_2 <= -1e-278)
		tmp = t_2;
	elseif (t_2 <= 2e-247)
		tmp = t_1;
	elseif (t_2 <= 5e+307)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(b / z), $MachinePrecision] + N[(t * N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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 * t), $MachinePrecision] / c), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$2, -1e-278], t$95$2, If[LessEqual[t$95$2, 2e-247], t$95$1, If[LessEqual[t$95$2, 5e+307], t$95$2, t$95$1]]]]]]

Target

Original	20.8
Target	14.3
Herbie	9.2

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

Derivation

1. Split input into 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

   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 49.3

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

      [Start] 64.0	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [=>] 64.0	\[ \frac{\left(\color{blue}{y \cdot \left(x \cdot 9\right)} - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      rational_best-simplify-44 [=>] 63.7	\[ \frac{\left(\color{blue}{x \cdot \left(y \cdot 9\right)} - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [=>] 63.7	\[ \frac{\left(x \cdot \color{blue}{\left(9 \cdot y\right)} - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [=>] 63.7	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{a \cdot \left(\left(z \cdot 4\right) \cdot t\right)}\right) + b}{z \cdot c} \]
      rational_best-simplify-44 [=>] 49.6	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(z \cdot 4\right) \cdot \left(a \cdot t\right)}\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [=>] 49.6	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{\left(a \cdot t\right) \cdot \left(z \cdot 4\right)}\right) + b}{z \cdot c} \]
      rational_best-simplify-44 [=>] 49.3	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - \color{blue}{z \cdot \left(\left(a \cdot t\right) \cdot 4\right)}\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [<=] 49.3	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - z \cdot \color{blue}{\left(4 \cdot \left(a \cdot t\right)\right)}\right) + b}{z \cdot c} \]
      rational_best-simplify-2 [=>] 49.3	\[ \frac{\left(x \cdot \left(9 \cdot y\right) - z \cdot \left(4 \cdot \color{blue}{\left(t \cdot a\right)}\right)\right) + b}{z \cdot c} \]

   3. Taylor expanded in z around inf 34.3

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

   4. Simplified 34.3

      \[\leadsto \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]
      Proof

      [Start] 34.3	\[ -4 \cdot \frac{a \cdot t}{c} \]
      rational_best-simplify-2 [=>] 34.3	\[ \color{blue}{\frac{a \cdot t}{c} \cdot -4} \]

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -9.99999999999999938e-279 or 2e-247 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 5e307

   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 -9.99999999999999938e-279 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 2e-247 or 5e307 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 52.1

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

   2. Taylor expanded in x around 0 52.3

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

   3. Simplified 52.4

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

      [Start] 52.3	\[ \frac{b - 4 \cdot \left(a \cdot \left(t \cdot z\right)\right)}{c \cdot z} \]
      rational_best-simplify-44 [=>] 52.4	\[ \frac{b - \color{blue}{a \cdot \left(4 \cdot \left(t \cdot z\right)\right)}}{c \cdot z} \]
      rational_best-simplify-2 [<=] 52.4	\[ \frac{b - a \cdot \left(4 \cdot \left(t \cdot z\right)\right)}{\color{blue}{z \cdot c}} \]

   4. Taylor expanded in b around 0 26.6

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

   5. Taylor expanded in c around 0 20.9

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

   6. Simplified 20.9

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

      [Start] 20.9	\[ \frac{\frac{b}{z} + -4 \cdot \left(a \cdot t\right)}{c} \]
      rational_best-simplify-2 [=>] 20.9	\[ \frac{\frac{b}{z} + -4 \cdot \color{blue}{\left(t \cdot a\right)}}{c} \]
      rational_best-simplify-44 [=>] 20.9	\[ \frac{\frac{b}{z} + \color{blue}{t \cdot \left(-4 \cdot a\right)}}{c} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\frac{a \cdot t}{c} \cdot -4\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.8
Cost	1892

\[\begin{array}{l} t_1 := \frac{a \cdot t}{c} \cdot -4\\ t_2 := \frac{\frac{b}{z}}{c}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-289}:\\ \;\;\;\;9 \cdot \frac{y \cdot x}{z \cdot c}\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-156}:\\ \;\;\;\;\frac{9 \cdot \left(y \cdot x\right)}{z \cdot c}\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 12600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+48}:\\ \;\;\;\;\frac{y \cdot \left(9 \cdot x\right)}{c \cdot z}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+268}:\\ \;\;\;\;\frac{t \cdot \left(-4 \cdot \left(z \cdot a\right)\right)}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.8
Cost	1880

\[\begin{array}{l} t_1 := \frac{b}{c \cdot z}\\ t_2 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-85}:\\ \;\;\;\;t_1 + 9 \cdot \frac{y \cdot x}{c \cdot z}\\ \mathbf{elif}\;a \leq 13500:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+217}:\\ \;\;\;\;t_1 + -4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{+268}:\\ \;\;\;\;\frac{9 \cdot \left(y \cdot x\right) - t \cdot \left(z \cdot \left(4 \cdot a\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	23.7
Cost	1624

\[\begin{array}{l} t_1 := \frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ t_2 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;a \leq -2.2 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.52 \cdot 10^{-86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 14200:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{+270}:\\ \;\;\;\;\frac{b - a \cdot \left(4 \cdot \left(t \cdot z\right)\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	36.2
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{z}}{c}\\ t_2 := \frac{a \cdot t}{c} \cdot -4\\ t_3 := 9 \cdot \frac{y \cdot x}{z \cdot c}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-247}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-30}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	36.1
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{z}}{c}\\ t_2 := \frac{a \cdot t}{c} \cdot -4\\ t_3 := \frac{9 \cdot \left(y \cdot x\right)}{z \cdot c}\\ \mathbf{if}\;z \leq -6 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-229}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-242}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;9 \cdot \frac{y \cdot x}{z \cdot c}\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{+98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	19.3
Cost	1488

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -1.28 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.26 \cdot 10^{-71}:\\ \;\;\;\;\frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-134}:\\ \;\;\;\;\frac{b - t \cdot \left(z \cdot \left(4 \cdot a\right)\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c \cdot z} + 9 \cdot \frac{y \cdot x}{c \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	13.8
Cost	1480

\[\begin{array}{l} t_1 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(x \cdot \left(9 \cdot y\right) - z \cdot \left(4 \cdot \left(t \cdot a\right)\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	23.3
Cost	1232

\[\begin{array}{l} t_1 := \frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ t_2 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 13000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	19.3
Cost	1232

\[\begin{array}{l} t_1 := \frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ t_2 := \frac{\frac{b}{z} + t \cdot \left(-4 \cdot a\right)}{c}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2.95 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.25 \cdot 10^{-134}:\\ \;\;\;\;\frac{b - t \cdot \left(z \cdot \left(4 \cdot a\right)\right)}{z \cdot c}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	35.0
Cost	976

\[\begin{array}{l} t_1 := \frac{a \cdot t}{c} \cdot -4\\ \mathbf{if}\;z \leq -7.4 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.1
Cost	968

\[\begin{array}{l} t_1 := \frac{a \cdot t}{c} \cdot -4\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+135}:\\ \;\;\;\;\frac{y \cdot \left(9 \cdot x\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	43.3
Cost	320

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

Reproduce

herbie shell --seed 2023092 
(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)))