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

Average Error: 20.4 → 6.3

Time: 31.1s

Precision: binary64

Cost: 11340

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

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{\frac{9 \cdot y}{\frac{z}{x}} + t_1}{c}\\ \mathbf{elif}\;t_2 \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-303}:\\ \;\;\;\;\frac{t_1 + \frac{1}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}}}{c}\\ \mathbf{elif}\;t_2 \leq 10^{+303}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

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 (* t (* a -4.0)))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_2 (- INFINITY))
     (/ (+ (/ (* 9.0 y) (/ z x)) t_1) c)
     (if (<= t_2 -4e-280)
       t_2
       (if (<= t_2 5e-303)
         (/ (+ t_1 (/ 1.0 (/ z (fma x (* 9.0 y) b)))) c)
         (if (<= t_2 1e+303) t_2 (* -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 = t * (a * -4.0);
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (((9.0 * y) / (z / x)) + t_1) / c;
	} else if (t_2 <= -4e-280) {
		tmp = t_2;
	} else if (t_2 <= 5e-303) {
		tmp = (t_1 + (1.0 / (z / fma(x, (9.0 * y), b)))) / c;
	} else if (t_2 <= 1e+303) {
		tmp = t_2;
	} 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(t * Float64(a * -4.0))
	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(Float64(9.0 * y) / Float64(z / x)) + t_1) / c);
	elseif (t_2 <= -4e-280)
		tmp = t_2;
	elseif (t_2 <= 5e-303)
		tmp = Float64(Float64(t_1 + Float64(1.0 / Float64(z / fma(x, Float64(9.0 * y), b)))) / c);
	elseif (t_2 <= 1e+303)
		tmp = t_2;
	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[(t * N[(a * -4.0), $MachinePrecision]), $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[(N[(9.0 * y), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, -4e-280], t$95$2, If[LessEqual[t$95$2, 5e-303], N[(N[(t$95$1 + N[(1.0 / N[(z / N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, 1e+303], t$95$2, N[(-4.0 * N[(a / N[(c / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	20.4
Target	14.7
Herbie	6.3

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

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 26.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z} + t \cdot \left(a \cdot -4\right)}{c}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (fma.f64 (*.f64 x 9) y b) z) (*.f64 t (*.f64 a -4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 x 9) y) b)) z) (*.f64 t (*.f64 a -4))) c): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 (*.f64 x 9) y))) z) (*.f64 t (*.f64 a -4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 t (*.f64 a 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 t) (*.f64 a 4)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (*.f64 a 4)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t a) 4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 t a)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (Rewrite<= metadata-eval (/.f64 4 1)) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (/.f64 4 (Rewrite<= *-inverses_binary64 (/.f64 z z))) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 4 z) z)) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z 4)) z) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 z 4) (/.f64 z (*.f64 t a))))) c): 17 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z 4) (*.f64 t a)) z))) c): 27 points increase in error, 14 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 10 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 3 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 b (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)))) z) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b)) z) c): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))): 47 points increase in error, 53 points decrease in error

   3. Taylor expanded in x around inf 30.9

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

   4. Simplified 22.9

      \[\leadsto \frac{\color{blue}{\frac{9 \cdot y}{\frac{z}{x}}} + t \cdot \left(a \cdot -4\right)}{c} \]
      Proof
      (/.f64 (*.f64 9 y) (/.f64 z x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 9 (/.f64 y (/.f64 z x)))): 30 points increase in error, 34 points decrease in error
      (*.f64 9 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) z))): 50 points increase in error, 38 points decrease in error

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -3.9999999999999998e-280 or 4.9999999999999998e-303 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 1e303

   1. Initial program 0.7

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

   if -3.9999999999999998e-280 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 4.9999999999999998e-303

   1. Initial program 35.6

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x \cdot 9, y, b\right)}{z} + t \cdot \left(a \cdot -4\right)}{c}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (fma.f64 (*.f64 x 9) y b) z) (*.f64 t (*.f64 a -4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 x 9) y) b)) z) (*.f64 t (*.f64 a -4))) c): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 (*.f64 x 9) y))) z) (*.f64 t (*.f64 a -4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 t (*.f64 a 4))))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 t) (*.f64 a 4)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 t (*.f64 a 4)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t a) 4))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 t a)))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (Rewrite<= metadata-eval (/.f64 4 1)) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (/.f64 4 (Rewrite<= *-inverses_binary64 (/.f64 z z))) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 4 z) z)) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (*.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 z 4)) z) (*.f64 t a))) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 z 4) (/.f64 z (*.f64 t a))))) c): 17 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 z 4) (*.f64 t a)) z))) c): 27 points increase in error, 14 points decrease in error
      (/.f64 (-.f64 (/.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) z) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 10 points increase in error, 3 points decrease in error
      (/.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (+.f64 b (*.f64 (*.f64 x 9) y)) (*.f64 (*.f64 (*.f64 z 4) t) a)) z)) c): 3 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= associate-+r-_binary64 (+.f64 b (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)))) z) c): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b)) z) c): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))): 47 points increase in error, 53 points decrease in error

   3. Applied egg-rr 0.4

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

   4. Applied egg-rr 0.4

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

   if 1e303 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 62.8

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

   2. Simplified 55.8

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(9 \cdot y\right) - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}} \]
      Proof
      (/.f64 (+.f64 (-.f64 (*.f64 x (*.f64 9 y)) (*.f64 (*.f64 z 4) (*.f64 t a))) b) (*.f64 z c)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x 9) y)) (*.f64 (*.f64 z 4) (*.f64 t a))) b) (*.f64 z c)): 6 points increase in error, 8 points decrease in error
      (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 z 4) t) a))) b) (*.f64 z c)): 12 points increase in error, 11 points decrease in error

   3. Taylor expanded in z around inf 30.8

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

   4. Simplified 22.8

      \[\leadsto \color{blue}{\frac{a}{\frac{c}{t}} \cdot -4} \]
      Proof
      (*.f64 (/.f64 a (/.f64 c t)) -4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 a t) c)) -4): 52 points increase in error, 44 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -4 (/.f64 (*.f64 a t) c))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\frac{\frac{9 \cdot y}{\frac{z}{x}} + t \cdot \left(a \cdot -4\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5 \cdot 10^{-303}:\\ \;\;\;\;\frac{t \cdot \left(a \cdot -4\right) + \frac{1}{\frac{z}{\mathsf{fma}\left(x, 9 \cdot y, b\right)}}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 10^{+303}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.8
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_2 := \frac{\frac{9 \cdot y}{\frac{z}{x}} + t \cdot \left(a \cdot -4\right)}{c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 2
Error	6.3
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{9 \cdot y}{\frac{z}{x}} + t \cdot \left(a \cdot -4\right)}{c}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-314}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b + \left(x \cdot \left(9 \cdot y\right) + z \cdot \left(a \cdot \left(t \cdot -4\right)\right)\right)}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 3
Error	6.3
Cost	6352

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{9 \cdot y}{\frac{z}{x}} + t \cdot \left(a \cdot -4\right)}{c}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{b + \left(x \cdot \left(9 \cdot y\right) + z \cdot \left(a \cdot \left(t \cdot -4\right)\right)\right)}{z} \cdot \frac{1}{c}\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Alternative 4
Error	39.4
Cost	2292

\[\begin{array}{l} t_1 := 9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ t_2 := -4 \cdot \frac{a}{\frac{c}{t}}\\ t_3 := \frac{-4 \cdot \left(t \cdot a\right)}{c}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{+171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{+135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-288}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-266}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-189}:\\ \;\;\;\;\frac{1}{c} \cdot \frac{b}{z}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-111}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z \cdot c}{x}}\\ \end{array} \]

Alternative 5
Error	28.7
Cost	2288

\[\begin{array}{l} t_1 := b + 9 \cdot \left(x \cdot y\right)\\ t_2 := \frac{t_1}{z \cdot c}\\ t_3 := \frac{t \cdot \left(a \cdot -4\right) + \frac{b}{z}}{c}\\ \mathbf{if}\;x \leq -9 \cdot 10^{+220}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{+172}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{t_1}{z}}{c}\\ \mathbf{elif}\;x \leq -4.3 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{+66}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z \cdot c}{x}}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+45}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-225}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot y}{c}}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Error	28.8
Cost	2288

\[\begin{array}{l} t_1 := b + 9 \cdot \left(x \cdot y\right)\\ t_2 := \frac{t_1}{z \cdot c}\\ t_3 := t \cdot \left(a \cdot -4\right)\\ t_4 := \frac{t_3 + \frac{b}{z}}{c}\\ t_5 := \frac{t_3 + \frac{1}{\frac{z}{b}}}{c}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+219}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{+171}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+133}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{t_1}{z}}{c}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+66}:\\ \;\;\;\;9 \cdot \frac{y}{\frac{z \cdot c}{x}}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+43}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.9 \cdot 10^{-32}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-173}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-225}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-110}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot y}{c}}{\frac{z}{x}}\\ \end{array} \]

Alternative 7
Error	24.1
Cost	2280

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := \frac{\frac{9 \cdot y}{\frac{z}{x}} + t_1}{c}\\ t_3 := \frac{t_1 + \frac{b}{z}}{c}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-232}:\\ \;\;\;\;\frac{b + t \cdot \left(a \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+186}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+304}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{z \cdot c}\right)\\ \end{array} \]

Alternative 8
Error	24.0
Cost	2280

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := \frac{\frac{9 \cdot y}{\frac{z}{x}} + t_1}{c}\\ t_3 := \frac{t_1 + \frac{b}{z}}{c}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-178}:\\ \;\;\;\;\frac{t_1 + \frac{\left(x \cdot 9\right) \cdot y}{z}}{c}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-231}:\\ \;\;\;\;\frac{b + t \cdot \left(a \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-10}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{elif}\;y \leq 0.18:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+186}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+304}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(9 \cdot \frac{x}{z \cdot c}\right)\\ \end{array} \]

Alternative 9
Error	29.4
Cost	2156

\[\begin{array}{l} t_1 := \frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ t_2 := t \cdot \left(a \cdot -4\right)\\ t_3 := \frac{t_2 + \frac{b}{z}}{c}\\ t_4 := \frac{t_2 + \frac{1}{\frac{z}{b}}}{c}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{+220}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -2.05 \cdot 10^{+172}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{+133}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{b}{z} + 9 \cdot \frac{x \cdot y}{z}}{c}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+83}:\\ \;\;\;\;-4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{elif}\;x \leq -9.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{\frac{c}{x}} \cdot \frac{9}{z}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-225}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot y}{c}}{\frac{z}{x}}\\ \end{array} \]

Alternative 10
Error	20.6
Cost	2140

\[\begin{array}{l} t_1 := t \cdot \left(a \cdot -4\right)\\ t_2 := \frac{\frac{9 \cdot y}{\frac{z}{x}} + t_1}{c}\\ t_3 := \frac{t_1 + \frac{b}{z}}{c}\\ t_4 := \frac{b + \left(x \cdot \left(9 \cdot y\right) + \left(t \cdot a\right) \cdot \left(z \cdot -4\right)\right)}{z \cdot c}\\ \mathbf{if}\;y \leq -9 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-268}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+140}:\\ \;\;\;\;\frac{t_1 + \frac{\left(x \cdot 9\right) \cdot y}{z}}{c}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+186}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+213}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 11
Error	28.3
Cost	1628

\[\begin{array}{l} t_1 := \frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ t_2 := \frac{t \cdot \left(a \cdot -4\right) + \frac{b}{z}}{c}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{+171}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-225}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-266}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{9 \cdot y}{c}}{\frac{z}{x}}\\ \end{array} \]

Alternative 12
Error	34.4
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ t_3 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-274}:\\ \;\;\;\;9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 820:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	34.4
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ t_3 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-272}:\\ \;\;\;\;9 \cdot \frac{\frac{x}{c}}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 950:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 14
Error	34.5
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ t_3 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\frac{c}{9}}\\ \mathbf{elif}\;a \leq 7.6 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 880:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	34.5
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ t_3 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -1.95 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-274}:\\ \;\;\;\;\frac{y}{\frac{c}{x}} \cdot \frac{9}{z}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6800000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 16
Error	34.4
Cost	1504

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := 9 \cdot \frac{x \cdot y}{z \cdot c}\\ t_3 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -7 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-274}:\\ \;\;\;\;\frac{9 \cdot y}{c \cdot \frac{z}{x}}\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-171}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{b}{c}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 165000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 17
Error	34.4
Cost	976

\[\begin{array}{l} t_1 := \frac{\frac{b}{c}}{z}\\ t_2 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-273}:\\ \;\;\;\;9 \cdot \left(\frac{x}{c} \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;a \leq 780:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	25.3
Cost	968

\[\begin{array}{l} t_1 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{b + 9 \cdot \left(x \cdot y\right)}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	34.3
Cost	712

\[\begin{array}{l} t_1 := -4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 600000:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	34.1
Cost	712

\[\begin{array}{l} t_1 := -4 \cdot \frac{a}{\frac{c}{t}}\\ \mathbf{if}\;a \leq -9.6 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 850000000:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	42.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;c \leq -1.08 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{b}{z}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 22
Error	43.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{b}{c}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{z \cdot c}\\ \end{array} \]

Alternative 23
Error	43.4
Cost	320

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

Reproduce

herbie shell --seed 2022331 
(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)))