quad2p (problem 3.2.1, positive)

Percentage Accurate: 38.9% → 69.3%

Time: 21.8s

Precision: binary64

Cost: 49424

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{-3}\right)}^{3} \cdot 0.001953125\right)}^{0.3333333333333333} + \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)))
   (if (<= t_0 (- INFINITY))
     (/ (* b_2 -2.0) a)
     (if (<= t_0 -2e-253)
       t_0
       (if (<= t_0 0.0)
         (* -0.5 (/ c b_2))
         (if (<= t_0 2e+234)
           t_0
           (+
            (* -2.0 (/ b_2 a))
            (+
             (pow
              (* (pow (* (* a (* c c)) (pow b_2 -3.0)) 3.0) 0.001953125)
              0.3333333333333333)
             (* (/ c b_2) 0.5)))))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double t_0 = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (b_2 * -2.0) / a;
	} else if (t_0 <= -2e-253) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -0.5 * (c / b_2);
	} else if (t_0 <= 2e+234) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (pow((pow(((a * (c * c)) * pow(b_2, -3.0)), 3.0) * 0.001953125), 0.3333333333333333) + ((c / b_2) * 0.5));
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double t_0 = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (b_2 * -2.0) / a;
	} else if (t_0 <= -2e-253) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = -0.5 * (c / b_2);
	} else if (t_0 <= 2e+234) {
		tmp = t_0;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (Math.pow((Math.pow(((a * (c * c)) * Math.pow(b_2, -3.0)), 3.0) * 0.001953125), 0.3333333333333333) + ((c / b_2) * 0.5));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	t_0 = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (b_2 * -2.0) / a
	elif t_0 <= -2e-253:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = -0.5 * (c / b_2)
	elif t_0 <= 2e+234:
		tmp = t_0
	else:
		tmp = (-2.0 * (b_2 / a)) + (math.pow((math.pow(((a * (c * c)) * math.pow(b_2, -3.0)), 3.0) * 0.001953125), 0.3333333333333333) + ((c / b_2) * 0.5))
	return tmp

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (t_0 <= -2e-253)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (t_0 <= 2e+234)
		tmp = t_0;
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64((Float64((Float64(Float64(a * Float64(c * c)) * (b_2 ^ -3.0)) ^ 3.0) * 0.001953125) ^ 0.3333333333333333) + Float64(Float64(c / b_2) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	t_0 = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (b_2 * -2.0) / a;
	elseif (t_0 <= -2e-253)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = -0.5 * (c / b_2);
	elseif (t_0 <= 2e+234)
		tmp = t_0;
	else
		tmp = (-2.0 * (b_2 / a)) + ((((((a * (c * c)) * (b_2 ^ -3.0)) ^ 3.0) * 0.001953125) ^ 0.3333333333333333) + ((c / b_2) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$0, -2e-253], t$95$0, If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+234], t$95$0, N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[Power[N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[Power[b$95$2, -3.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * 0.001953125), $MachinePrecision], 0.3333333333333333], $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -inf.0

   1. Initial program 25.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 25.6%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 25.6%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 25.6%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 62.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]

   4. Simplified 62.3%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Step-by-step derivation

      [Start] 62.3%	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 62.3%	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -inf.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -2.0000000000000001e-253 or 0.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 2.00000000000000004e234

   1. Initial program 87.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 87.1%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 87.1%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 87.1%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   if -2.0000000000000001e-253 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 0.0

   1. Initial program 17.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 17.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 17.3%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 17.3%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 17.3%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 100.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if 2.00000000000000004e234 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a)

   1. Initial program 9.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Simplified 9.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
      Step-by-step derivation

      [Start] 9.4%	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 9.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 9.4%	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 34.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right)} \]

   4. Applied egg-rr 54.7%

      \[\leadsto -2 \cdot \frac{b_2}{a} + \left(\color{blue}{{\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{-3}\right)}^{3} \cdot 0.001953125\right)}^{0.3333333333333333}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      Step-by-step derivation

      [Start] 34.6%	\[ -2 \cdot \frac{b_2}{a} + \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      add-cbrt-cube [=>] 34.6%	\[ -2 \cdot \frac{b_2}{a} + \left(\color{blue}{\sqrt[3]{\left(\left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right) \cdot \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)\right) \cdot \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      pow1/3 [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left(\color{blue}{{\left(\left(\left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right) \cdot \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)\right) \cdot \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)\right)}^{0.3333333333333333}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      pow3 [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\color{blue}{\left({\left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)}^{3}\right)}}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      *-commutative [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\color{blue}{\left(\frac{{c}^{2} \cdot a}{{b_2}^{3}} \cdot 0.125\right)}}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      unpow-prod-down [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\color{blue}{\left({\left(\frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)}^{3} \cdot {0.125}^{3}\right)}}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      div-inv [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\color{blue}{\left(\left({c}^{2} \cdot a\right) \cdot \frac{1}{{b_2}^{3}}\right)}}^{3} \cdot {0.125}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      *-commutative [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\left(\color{blue}{\left(a \cdot {c}^{2}\right)} \cdot \frac{1}{{b_2}^{3}}\right)}^{3} \cdot {0.125}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      unpow2 [=>] 53.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \frac{1}{{b_2}^{3}}\right)}^{3} \cdot {0.125}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      pow-flip [=>] 54.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot \color{blue}{{b_2}^{\left(-3\right)}}\right)}^{3} \cdot {0.125}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      metadata-eval [=>] 54.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{\color{blue}{-3}}\right)}^{3} \cdot {0.125}^{3}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]
      metadata-eval [=>] 54.7%	\[ -2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{-3}\right)}^{3} \cdot \color{blue}{0.001953125}\right)}^{0.3333333333333333} + 0.5 \cdot \frac{c}{b_2}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \leq -\infty:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \leq -2 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a} \leq 2 \cdot 10^{+234}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{-3}\right)}^{3} \cdot 0.001953125\right)}^{0.3333333333333333} + \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.3%
Cost	49424

\[\begin{array}{l} t_0 := \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left({\left({\left(\left(a \cdot \left(c \cdot c\right)\right) \cdot {b_2}^{-3}\right)}^{3} \cdot 0.001953125\right)}^{0.3333333333333333} + \frac{c}{b_2} \cdot 0.5\right)\\ \end{array} \]

Alternative 2
Accuracy	65.5%
Cost	20232

\[\begin{array}{l} t_0 := \frac{c}{\frac{\frac{{b_2}^{3}}{c}}{a}}\\ t_1 := a \cdot \left(-c\right)\\ t_2 := a \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(0.5 \cdot \sqrt[3]{t_2 \cdot \left(t_2 \cdot t_2\right)} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.75 \cdot 10^{-201}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, t_1\right)\right)}^{0.25}\right)}^{2} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -4.5 \cdot 10^{-285}:\\ \;\;\;\;-0.125 \cdot t_0\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{{\left({t_1}^{0.25}\right)}^{2} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, t_0, -0.5 \cdot \frac{c}{b_2}\right)\\ \end{array} \]

Alternative 3
Accuracy	64.7%
Cost	14224

\[\begin{array}{l} t_0 := \frac{c}{\frac{\frac{{b_2}^{3}}{c}}{a}}\\ t_1 := a \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.76 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(0.5 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -8.8 \cdot 10^{-290}:\\ \;\;\;\;-0.125 \cdot t_0\\ \mathbf{elif}\;b_2 \leq 2.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{{\left({\left(a \cdot \left(-c\right)\right)}^{0.25}\right)}^{2} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, t_0, -0.5 \cdot \frac{c}{b_2}\right)\\ \end{array} \]

Alternative 4
Accuracy	64.5%
Cost	13968

\[\begin{array}{l} t_0 := \frac{{b_2}^{3}}{c}\\ t_1 := a \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(0.5 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.35 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.15 \cdot 10^{-286}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{t_0}{a}}\\ \mathbf{elif}\;b_2 \leq 3.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{{\left({\left(a \cdot \left(-c\right)\right)}^{0.25}\right)}^{2} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + -0.125 \cdot \left(a \cdot \frac{c}{t_0}\right)\\ \end{array} \]

Alternative 5
Accuracy	61.5%
Cost	8336

\[\begin{array}{l} t_0 := \frac{{b_2}^{3}}{c}\\ t_1 := -2 \cdot \frac{b_2}{a}\\ t_2 := \frac{c}{b_2} \cdot 0.5\\ \mathbf{if}\;b_2 \leq -3.4 \cdot 10^{+74}:\\ \;\;\;\;t_1 + t_2\\ \mathbf{elif}\;b_2 \leq -2.4 \cdot 10^{-200}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5.9 \cdot 10^{-241}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{t_0}{a}}\\ \mathbf{elif}\;b_2 \leq 3 \cdot 10^{-141}:\\ \;\;\;\;t_1 + \left(t_2 + \left(c \cdot \left(c \cdot {b_2}^{-3}\right)\right) \cdot \left(a \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + -0.125 \cdot \left(a \cdot \frac{c}{t_0}\right)\\ \end{array} \]

Alternative 6
Accuracy	61.1%
Cost	8336

\[\begin{array}{l} t_0 := \frac{{b_2}^{3}}{c}\\ t_1 := a \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -3.05 \cdot 10^{+74}:\\ \;\;\;\;\frac{\left(0.5 \cdot \sqrt[3]{t_1 \cdot \left(t_1 \cdot t_1\right)} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.65 \cdot 10^{-227}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{t_0}{a}}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-142}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \left(\frac{c}{b_2} \cdot 0.5 + \left(c \cdot \left(c \cdot {b_2}^{-3}\right)\right) \cdot \left(a \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + -0.125 \cdot \left(a \cdot \frac{c}{t_0}\right)\\ \end{array} \]

Alternative 7
Accuracy	57.6%
Cost	7704

\[\begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ t_1 := \frac{c}{\frac{b_2}{a}}\\ \mathbf{if}\;b_2 \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(0.5 \cdot t_1 - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -3.05 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -7.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \leq -1.15 \cdot 10^{-208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -3.65 \cdot 10^{-301}:\\ \;\;\;\;\frac{-0.5 \cdot t_1}{a}\\ \mathbf{elif}\;b_2 \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 8
Accuracy	56.3%
Cost	7700

\[\begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{if}\;b_2 \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.1 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.25 \cdot 10^{-146}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \leq -7.5 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 4 \cdot 10^{-155}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{\frac{{b_2}^{3}}{c}}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 9
Accuracy	56.3%
Cost	7700

\[\begin{array}{l} t_0 := \sqrt{a \cdot \left(-c\right)}\\ \mathbf{if}\;b_2 \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -3.1 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_0}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -8.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \leq -3.4 \cdot 10^{-199}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.8 \cdot 10^{-155}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{\frac{{b_2}^{3}}{c}}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 10
Accuracy	61.7%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3 \cdot 10^{+74}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq -1.48 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2} + -0.125 \cdot \left(a \cdot \frac{c}{\frac{{b_2}^{3}}{c}}\right)\\ \end{array} \]

Alternative 11
Accuracy	61.3%
Cost	7436

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq -1.3 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.5 \cdot 10^{-155}:\\ \;\;\;\;-0.125 \cdot \frac{c}{\frac{\frac{{b_2}^{3}}{c}}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 12
Accuracy	56.4%
Cost	1100

\[\begin{array}{l} t_0 := \frac{c}{\frac{b_2}{a}}\\ \mathbf{if}\;b_2 \leq -2.45 \cdot 10^{-201}:\\ \;\;\;\;\frac{\left(0.5 \cdot t_0 - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\frac{-0.5 \cdot t_0}{a}\\ \mathbf{elif}\;b_2 \leq 6 \cdot 10^{-142}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b_2}\right) - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 13
Accuracy	55.5%
Cost	964

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-201}:\\ \;\;\;\;\frac{b_2 \cdot -2 + 0.5 \cdot \frac{a \cdot c}{b_2}}{a}\\ \mathbf{elif}\;b_2 \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 14
Accuracy	54.6%
Cost	840

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{-160}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 15
Accuracy	55.5%
Cost	840

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.4 \cdot 10^{-200}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 4.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c}{\frac{b_2}{a}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 16
Accuracy	38.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.35 \cdot 10^{-198}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 17
Accuracy	53.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 18
Accuracy	18.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 19
Accuracy	9.2%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023255 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))