jeff quadratic root 2

Average Error: 19.8 → 6.6

Time: 20.6s

Precision: binary64

Cost: 38052

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

↓

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ t_2 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+265}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (- b) b))
        (t_1 (if (>= b 0.0) (/ (* 2.0 c) t_0) (/ (- b) a)))
        (t_2 (sqrt (- (* b b) (* c (* 4.0 a)))))
        (t_3
         (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_2)) (/ (- t_2 b) (* 2.0 a)))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -5e-274)
       t_3
       (if (<= t_3 0.0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- (* 2.0 (/ c (/ b a))) b) b))
           (/ t_0 (* 2.0 a)))
         (if (<= t_3 5e+265) t_3 t_1))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	} else {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_0;
	} else {
		tmp = -b / a;
	}
	double t_1 = tmp;
	double t_2 = sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_2);
	} else {
		tmp_1 = (t_2 - b) / (2.0 * a);
	}
	double t_3 = tmp_1;
	double tmp_2;
	if (t_3 <= -((double) INFINITY)) {
		tmp_2 = t_1;
	} else if (t_3 <= -5e-274) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (((2.0 * (c / (b / a))) - b) - b);
		} else {
			tmp_3 = t_0 / (2.0 * a);
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= 5e+265) {
		tmp_2 = t_3;
	} else {
		tmp_2 = t_1;
	}
	return tmp_2;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - Math.sqrt(((b * b) - ((4.0 * a) * c))));
	} else {
		tmp = (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	}
	return tmp;
}

↓

public static double code(double a, double b, double c) {
	double t_0 = -b - b;
	double tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / t_0;
	} else {
		tmp = -b / a;
	}
	double t_1 = tmp;
	double t_2 = Math.sqrt(((b * b) - (c * (4.0 * a))));
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_2);
	} else {
		tmp_1 = (t_2 - b) / (2.0 * a);
	}
	double t_3 = tmp_1;
	double tmp_2;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp_2 = t_1;
	} else if (t_3 <= -5e-274) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (((2.0 * (c / (b / a))) - b) - b);
		} else {
			tmp_3 = t_0 / (2.0 * a);
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= 5e+265) {
		tmp_2 = t_3;
	} else {
		tmp_2 = t_1;
	}
	return tmp_2;
}

Python

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - math.sqrt(((b * b) - ((4.0 * a) * c))))
	else:
		tmp = (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
	return tmp

↓

def code(a, b, c):
	t_0 = -b - b
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / t_0
	else:
		tmp = -b / a
	t_1 = tmp
	t_2 = math.sqrt(((b * b) - (c * (4.0 * a))))
	tmp_1 = 0
	if b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - t_2)
	else:
		tmp_1 = (t_2 - b) / (2.0 * a)
	t_3 = tmp_1
	tmp_2 = 0
	if t_3 <= -math.inf:
		tmp_2 = t_1
	elif t_3 <= -5e-274:
		tmp_2 = t_3
	elif t_3 <= 0.0:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (((2.0 * (c / (b / a))) - b) - b)
		else:
			tmp_3 = t_0 / (2.0 * a)
		tmp_2 = tmp_3
	elif t_3 <= 5e+265:
		tmp_2 = t_3
	else:
		tmp_2 = t_1
	return tmp_2

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
	else
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	end
	return tmp
end

↓

function code(a, b, c)
	t_0 = Float64(Float64(-b) - b)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / t_0);
	else
		tmp = Float64(Float64(-b) / a);
	end
	t_1 = tmp
	t_2 = sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a))))
	tmp_1 = 0.0
	if (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_2));
	else
		tmp_1 = Float64(Float64(t_2 - b) / Float64(2.0 * a));
	end
	t_3 = tmp_1
	tmp_2 = 0.0
	if (t_3 <= Float64(-Inf))
		tmp_2 = t_1;
	elseif (t_3 <= -5e-274)
		tmp_2 = t_3;
	elseif (t_3 <= 0.0)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(2.0 * c) / Float64(Float64(Float64(2.0 * Float64(c / Float64(b / a))) - b) - b));
		else
			tmp_3 = Float64(t_0 / Float64(2.0 * a));
		end
		tmp_2 = tmp_3;
	elseif (t_3 <= 5e+265)
		tmp_2 = t_3;
	else
		tmp_2 = t_1;
	end
	return tmp_2
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c))));
	else
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	end
	tmp_2 = tmp;
end

↓

function tmp_5 = code(a, b, c)
	t_0 = -b - b;
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / t_0;
	else
		tmp = -b / a;
	end
	t_1 = tmp;
	t_2 = sqrt(((b * b) - (c * (4.0 * a))));
	tmp_2 = 0.0;
	if (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - t_2);
	else
		tmp_2 = (t_2 - b) / (2.0 * a);
	end
	t_3 = tmp_2;
	tmp_3 = 0.0;
	if (t_3 <= -Inf)
		tmp_3 = t_1;
	elseif (t_3 <= -5e-274)
		tmp_3 = t_3;
	elseif (t_3 <= 0.0)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (((2.0 * (c / (b / a))) - b) - b);
		else
			tmp_4 = t_0 / (2.0 * a);
		end
		tmp_3 = tmp_4;
	elseif (t_3 <= 5e+265)
		tmp_3 = t_3;
	else
		tmp_3 = t_1;
	end
	tmp_5 = tmp_3;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[((-b) - b), $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / t$95$0), $MachinePrecision], N[((-b) / a), $MachinePrecision]]}, Block[{t$95$2 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -5e-274], t$95$3, If[LessEqual[t$95$3, 0.0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$3, 5e+265], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -inf.0 or 5.0000000000000002e265 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)))

   1. Initial program 60.7

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in b around inf 58.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Taylor expanded in b around -inf 16.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -5e-274 or -0.0 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < 5.0000000000000002e265

   1. Initial program 2.6

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if -5e-274 < (if (>=.f64 b 0) (/.f64 (*.f64 2 c) (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) < -0.0

   1. Initial program 35.8

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   2. Taylor expanded in b around -inf 35.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

   3. Taylor expanded in b around inf 12.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

   4. Simplified 10.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\left(b + -2 \cdot \frac{c}{\frac{b}{a}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]
      Proof
      (+.f64 b (*.f64 -2 (/.f64 c (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
      (+.f64 b (*.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)))): 21 points increase in error, 13 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq -5 \cdot 10^{-274}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	7952

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2 \cdot c}{t_0}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{if}\;b \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	7.8
Cost	7952

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ t_2 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -33000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-291}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+144}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.1
Cost	7888

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{if}\;b \leq -9.2 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-8}{\frac{-4}{b}} \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.4
Cost	7632

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{2 \cdot c}{t_0}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-8}{\frac{-4}{b}} \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2 \cdot a}\\ \end{array} \]

Alternative 5
Error	18.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-78}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{-2}{b + \sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-8}{\frac{-4}{b}} \cdot \frac{-0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array} \]

Alternative 6
Error	22.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{a}{\frac{b}{c}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array} \]

Alternative 7
Error	22.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \frac{c}{\frac{b}{a}} - b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array} \]

Alternative 8
Error	22.8
Cost	644

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022296 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))