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

Average Error: 31.78% → 12.35%

Time: 33.4s

Precision: binary64

Cost: 16328

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) + \frac{b + \left(z \cdot a\right) \cdot \left(t \cdot -4\right)}{z \cdot c}\\ \mathbf{elif}\;t_1 \leq -5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, z \cdot \left(t \cdot -4\right), b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

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))))
   (if (<= t_1 (- INFINITY))
     (+ (* 9.0 (* (/ y c) (/ x z))) (/ (+ b (* (* z a) (* t -4.0))) (* z c)))
     (if (<= t_1 -5000.0)
       (/ (fma x (* 9.0 y) (fma a (* z (* t -4.0)) b)) (* z c))
       (if (<= t_1 2e-5)
         (/ (+ (* a (* t -4.0)) (/ (+ b (* x (* 9.0 y))) z)) c)
         (if (<= t_1 1e+304) t_1 (* -4.0 (/ a (/ c t)))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (9.0 * ((y / c) * (x / z))) + ((b + ((z * a) * (t * -4.0))) / (z * c));
	} else if (t_1 <= -5000.0) {
		tmp = fma(x, (9.0 * y), fma(a, (z * (t * -4.0)), b)) / (z * c);
	} else if (t_1 <= 2e-5) {
		tmp = ((a * (t * -4.0)) + ((b + (x * (9.0 * y))) / z)) / c;
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = -4.0 * (a / (c / t));
	}
	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))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(9.0 * Float64(Float64(y / c) * Float64(x / z))) + Float64(Float64(b + Float64(Float64(z * a) * Float64(t * -4.0))) / Float64(z * c)));
	elseif (t_1 <= -5000.0)
		tmp = Float64(fma(x, Float64(9.0 * y), fma(a, Float64(z * Float64(t * -4.0)), b)) / Float64(z * c));
	elseif (t_1 <= 2e-5)
		tmp = Float64(Float64(Float64(a * Float64(t * -4.0)) + Float64(Float64(b + Float64(x * Float64(9.0 * y))) / z)) / c);
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = Float64(-4.0 * Float64(a / Float64(c / t)));
	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]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(9.0 * N[(N[(y / c), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b + N[(N[(z * a), $MachinePrecision] * N[(t * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5000.0], N[(N[(x * N[(9.0 * y), $MachinePrecision] + N[(a * N[(z * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(N[(N[(a * N[(t * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b + N[(x * N[(9.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], t$95$1, N[(-4.0 * N[(a / N[(c / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	31.78%
Target	22.91%
Herbie	12.35%

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

Derivation

1. Split input into 5 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 100

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

   2. Applied egg-rr 76.72

      \[\leadsto \color{blue}{\frac{1}{z \cdot c} \cdot \left(\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b\right)} \]

   3. Applied egg-rr 76.53

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

   4. Simplified 68.56

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

      [Start] 76.53	\[ \frac{9 \cdot \left(y \cdot x\right)}{z \cdot c} - \frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c} \]
      associate-*r/ [<=] 76.53	\[ \color{blue}{9 \cdot \frac{y \cdot x}{z \cdot c}} - \frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c} \]
      *-commutative [=>] 76.53	\[ 9 \cdot \frac{y \cdot x}{\color{blue}{c \cdot z}} - \frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c} \]
      times-frac [=>] 66.8	\[ 9 \cdot \color{blue}{\left(\frac{y}{c} \cdot \frac{x}{z}\right)} - \frac{\mathsf{fma}\left(z, 4 \cdot \left(t \cdot a\right), -b\right)}{z \cdot c} \]
      fma-udef [=>] 66.8	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{z \cdot \left(4 \cdot \left(t \cdot a\right)\right) + \left(-b\right)}}{z \cdot c} \]
      unsub-neg [=>] 66.8	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{z \cdot \left(4 \cdot \left(t \cdot a\right)\right) - b}}{z \cdot c} \]
      *-commutative [=>] 66.8	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{\left(4 \cdot \left(t \cdot a\right)\right) \cdot z} - b}{z \cdot c} \]
      associate-*r* [=>] 66.98	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{\left(\left(4 \cdot t\right) \cdot a\right)} \cdot z - b}{z \cdot c} \]
      associate-*l* [=>] 68.56	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{\left(4 \cdot t\right) \cdot \left(a \cdot z\right)} - b}{z \cdot c} \]
      *-commutative [=>] 68.56	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\color{blue}{\left(t \cdot 4\right)} \cdot \left(a \cdot z\right) - b}{z \cdot c} \]
      *-commutative [=>] 68.56	\[ 9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) - \frac{\left(t \cdot 4\right) \cdot \left(a \cdot z\right) - b}{\color{blue}{c \cdot z}} \]

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -5e3

   1. Initial program 0.9

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

   2. Simplified 0.89

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

      [Start] 0.9	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-+l- [=>] 0.9	\[ \frac{\color{blue}{\left(x \cdot 9\right) \cdot y - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}}{z \cdot c} \]
      associate-*l* [=>] 0.89	\[ \frac{\color{blue}{x \cdot \left(9 \cdot y\right)} - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}{z \cdot c} \]
      fma-neg [=>] 0.89	\[ \frac{\color{blue}{\mathsf{fma}\left(x, 9 \cdot y, -\left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)\right)}}{z \cdot c} \]
      neg-sub0 [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{0 - \left(\left(\left(z \cdot 4\right) \cdot t\right) \cdot a - b\right)}\right)}{z \cdot c} \]
      associate-+l- [<=] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(0 - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}\right)}{z \cdot c} \]
      neg-sub0 [<=] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(-\left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right)} + b\right)}{z \cdot c} \]
      distribute-lft-neg-in [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\left(-\left(z \cdot 4\right) \cdot t\right) \cdot a} + b\right)}{z \cdot c} \]
      *-commutative [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{a \cdot \left(-\left(z \cdot 4\right) \cdot t\right)} + b\right)}{z \cdot c} \]
      fma-def [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \color{blue}{\mathsf{fma}\left(a, -\left(z \cdot 4\right) \cdot t, b\right)}\right)}{z \cdot c} \]
      associate-*l* [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, -\color{blue}{z \cdot \left(4 \cdot t\right)}, b\right)\right)}{z \cdot c} \]
      distribute-rgt-neg-in [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, \color{blue}{z \cdot \left(-4 \cdot t\right)}, b\right)\right)}{z \cdot c} \]
      *-commutative [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, z \cdot \left(-\color{blue}{t \cdot 4}\right), b\right)\right)}{z \cdot c} \]
      distribute-rgt-neg-in [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, z \cdot \color{blue}{\left(t \cdot \left(-4\right)\right)}, b\right)\right)}{z \cdot c} \]
      metadata-eval [=>] 0.89	\[ \frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, z \cdot \left(t \cdot \color{blue}{-4}\right), b\right)\right)}{z \cdot c} \]

   if -5e3 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 2.00000000000000016e-5

   1. Initial program 21.8

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

   2. Simplified 1.42

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

      [Start] 21.8	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 1.93	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   3. Applied egg-rr 1.42

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

   if 2.00000000000000016e-5 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 9.9999999999999994e303

   1. Initial program 0.95

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

   if 9.9999999999999994e303 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 98.57

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

   2. Simplified 42.63

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

      [Start] 98.57	\[ \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]
      associate-/r* [=>] 95.52	\[ \color{blue}{\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}} \]

   3. Taylor expanded in a around inf 48.87

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

   4. Simplified 37.12

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

      [Start] 48.87	\[ -4 \cdot \frac{a \cdot t}{c} \]
      associate-/l* [=>] 37.12	\[ -4 \cdot \color{blue}{\frac{a}{\frac{c}{t}}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 12.35

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) + \frac{b + \left(z \cdot a\right) \cdot \left(t \cdot -4\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq -5000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 9 \cdot y, \mathsf{fma}\left(a, z \cdot \left(t \cdot -4\right), b\right)\right)}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c} \leq 10^{+304}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.93%
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;t_1 \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 2
Error	13.45%
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y + a \cdot \left(t \cdot \left(z \cdot -4\right)\right)\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+255}:\\ \;\;\;\;9 \cdot \left(\frac{y}{c} \cdot \frac{x}{z}\right) + \frac{b + \left(z \cdot a\right) \cdot \left(t \cdot -4\right)}{z \cdot c}\\ \mathbf{elif}\;t_1 \leq -5000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + x \cdot \left(9 \cdot y\right)}{z}}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 3
Error	54.73%
Cost	2032

\[\begin{array}{l} t_1 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_2 := \frac{\frac{b}{c}}{z}\\ t_3 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ t_4 := \left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{if}\;b \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-136}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;b \leq -1.05 \cdot 10^{-177}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -2.05 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -3.4 \cdot 10^{-275}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -5.2 \cdot 10^{-301}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-205}:\\ \;\;\;\;9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right)\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-123}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-84}:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;b \leq 0.0014:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 4
Error	54.53%
Cost	2032

\[\begin{array}{l} t_1 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_2 := \frac{\frac{b}{c}}{z}\\ t_3 := \left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-177}:\\ \;\;\;\;9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;b \leq -1.15 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-270}:\\ \;\;\;\;\left(x \cdot \frac{y}{c}\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-301}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-205}:\\ \;\;\;\;9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right)\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-84}:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;b \leq 0.0016:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 5
Error	54.6%
Cost	2032

\[\begin{array}{l} t_1 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_2 := \frac{\frac{b}{c}}{z}\\ t_3 := \left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{if}\;b \leq -7 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.42 \cdot 10^{-136}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{c} \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;b \leq -2.7 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9.8 \cdot 10^{-273}:\\ \;\;\;\;\left(x \cdot \frac{y}{c}\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;b \leq -2 \cdot 10^{-301}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-204}:\\ \;\;\;\;9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right)\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{-121}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-84}:\\ \;\;\;\;9 \cdot \frac{y}{z \cdot \frac{c}{x}}\\ \mathbf{elif}\;b \leq 0.00155:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 6
Error	54.57%
Cost	2032

\[\begin{array}{l} t_1 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_2 := \frac{\frac{b}{c}}{z}\\ t_3 := \left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-135}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;b \leq -7.4 \cdot 10^{-179}:\\ \;\;\;\;\frac{x}{c} \cdot \frac{9 \cdot y}{z}\\ \mathbf{elif}\;b \leq -5.7 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.75 \cdot 10^{-270}:\\ \;\;\;\;\left(x \cdot \frac{y}{c}\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-203}:\\ \;\;\;\;9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right)\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-83}:\\ \;\;\;\;\frac{9}{z} \cdot \frac{x \cdot y}{c}\\ \mathbf{elif}\;b \leq 0.0016:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 10^{+165}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 7
Error	54.32%
Cost	1768

\[\begin{array}{l} t_1 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_2 := \frac{\frac{b}{c}}{z}\\ t_3 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{if}\;b \leq -5.5 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.26 \cdot 10^{-136}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-180}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-270}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-204}:\\ \;\;\;\;9 \cdot \left(y \cdot \frac{x}{z \cdot c}\right)\\ \mathbf{elif}\;b \leq 0.0016:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 8
Error	41.95%
Cost	1760

\[\begin{array}{l} t_1 := \frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+192}:\\ \;\;\;\;\frac{-4}{\frac{\frac{c}{t}}{a}}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{+78}:\\ \;\;\;\;\left(t \cdot a\right) \cdot \frac{-4}{c}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{+33}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;t \leq -0.0034:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-18}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array} \]

Alternative 9
Error	53.07%
Cost	1505

\[\begin{array}{l} t_1 := -4 \cdot \frac{t}{\frac{c}{a}}\\ t_2 := \frac{t \cdot \left(a \cdot -4\right)}{c}\\ t_3 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;b \leq -1.5 \cdot 10^{-136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8.8 \cdot 10^{-178}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -4.6 \cdot 10^{-238}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-269}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+108} \lor \neg \left(b \leq 1.6 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	31.19%
Cost	1488

\[\begin{array}{l} t_1 := \frac{a \cdot \left(t \cdot -4\right) + 9 \cdot \frac{x \cdot y}{z}}{c}\\ t_2 := -4 \cdot \frac{t \cdot a}{c} + \frac{b}{z \cdot c}\\ \mathbf{if}\;b \leq -5.2 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -2.2 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-276}:\\ \;\;\;\;\left(x \cdot \frac{y}{c}\right) \cdot \frac{9}{z}\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	14.35%
Cost	1481

\[\begin{array}{l} t_1 := x \cdot \left(9 \cdot y\right)\\ \mathbf{if}\;z \leq -2000000 \lor \neg \left(z \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{b + t_1}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(t_1 + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \end{array} \]

Alternative 12
Error	19.52%
Cost	1353

\[\begin{array}{l} t_1 := b + x \cdot \left(9 \cdot y\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-21} \lor \neg \left(z \leq 10^{-281}\right):\\ \;\;\;\;\frac{a \cdot \left(t \cdot -4\right) + \frac{t_1}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{z \cdot c}\\ \end{array} \]

Alternative 13
Error	55.14%
Cost	1241

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ t_2 := -4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -0.62:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-44} \lor \neg \left(t \leq 7.6 \cdot 10^{-223}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \end{array} \]

Alternative 14
Error	55.17%
Cost	1240

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ t_2 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -0.017:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-45}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-223}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	56.14%
Cost	1240

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ t_2 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -0.085:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-45}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array} \]

Alternative 16
Error	56.1%
Cost	1240

\[\begin{array}{l} t_1 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{+97}:\\ \;\;\;\;b \cdot \frac{\frac{1}{z}}{c}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -0.38:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-43}:\\ \;\;\;\;-4 \cdot \left(a \cdot \frac{t}{c}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{t}{\frac{c}{a}}\\ \end{array} \]

Alternative 17
Error	30.01%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-21} \lor \neg \left(z \leq 6.5 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{\frac{b}{z} + -4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + x \cdot \left(9 \cdot y\right)}{z \cdot c}\\ \end{array} \]

Alternative 18
Error	65.63%
Cost	585

\[\begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{-43} \lor \neg \left(c \leq 1.4 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \end{array} \]

Alternative 19
Error	67.78%
Cost	320

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

Alternative 20
Error	67.83%
Cost	320

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

Reproduce

herbie shell --seed 2023090 
(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)))