jeff quadratic root 1

Average Error: 20.1 → 8.2

Time: 12.5s

Precision: binary64

Cost: 38052

Math

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

↓

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t_1}\\ \end{array}\\ t_3 := -1 \cdot \frac{b}{a}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array}\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-145}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq 10^{+240}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (if (>= b 0.0) (/ (- (- b) b) (* a 2.0)) (* -1.0 (/ c b))))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_2
         (if (>= b 0.0)
           (/ (- (- b) t_1) (* 2.0 a))
           (/ (* 2.0 c) (+ (- b) t_1))))
        (t_3 (* -1.0 (/ b a))))
   (if (<= t_2 (- INFINITY))
     (if (>= b 0.0) t_3 t_3)
     (if (<= t_2 -2e-145)
       t_2
       (if (<= t_2 0.0) t_0 (if (<= t_2 1e+240) t_2 t_0))))))

C

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

↓

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = -1.0 * (c / b);
	}
	double t_0 = tmp;
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (-b - t_1) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_1);
	}
	double t_2 = tmp_1;
	double t_3 = -1.0 * (b / a);
	double tmp_3;
	if (t_2 <= -((double) INFINITY)) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_3;
		} else {
			tmp_4 = t_3;
		}
		tmp_3 = tmp_4;
	} else if (t_2 <= -2e-145) {
		tmp_3 = t_2;
	} else if (t_2 <= 0.0) {
		tmp_3 = t_0;
	} else if (t_2 <= 1e+240) {
		tmp_3 = t_2;
	} else {
		tmp_3 = t_0;
	}
	return tmp_3;
}

Java

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

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - b) / (a * 2.0);
	} else {
		tmp = -1.0 * (c / b);
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (-b - t_1) / (2.0 * a);
	} else {
		tmp_1 = (2.0 * c) / (-b + t_1);
	}
	double t_2 = tmp_1;
	double t_3 = -1.0 * (b / a);
	double tmp_3;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_3;
		} else {
			tmp_4 = t_3;
		}
		tmp_3 = tmp_4;
	} else if (t_2 <= -2e-145) {
		tmp_3 = t_2;
	} else if (t_2 <= 0.0) {
		tmp_3 = t_0;
	} else if (t_2 <= 1e+240) {
		tmp_3 = t_2;
	} else {
		tmp_3 = t_0;
	}
	return tmp_3;
}

Python

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

↓

def code(a, b, c):
	tmp = 0
	if b >= 0.0:
		tmp = (-b - b) / (a * 2.0)
	else:
		tmp = -1.0 * (c / b)
	t_0 = tmp
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp_1 = 0
	if b >= 0.0:
		tmp_1 = (-b - t_1) / (2.0 * a)
	else:
		tmp_1 = (2.0 * c) / (-b + t_1)
	t_2 = tmp_1
	t_3 = -1.0 * (b / a)
	tmp_3 = 0
	if t_2 <= -math.inf:
		tmp_4 = 0
		if b >= 0.0:
			tmp_4 = t_3
		else:
			tmp_4 = t_3
		tmp_3 = tmp_4
	elif t_2 <= -2e-145:
		tmp_3 = t_2
	elif t_2 <= 0.0:
		tmp_3 = t_0
	elif t_2 <= 1e+240:
		tmp_3 = t_2
	else:
		tmp_3 = t_0
	return tmp_3

Julia

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

↓

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp_1 = 0.0
	if (b >= 0.0)
		tmp_1 = Float64(Float64(Float64(-b) - t_1) / Float64(2.0 * a));
	else
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_1));
	end
	t_2 = tmp_1
	t_3 = Float64(-1.0 * Float64(b / a))
	tmp_3 = 0.0
	if (t_2 <= Float64(-Inf))
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_3;
		else
			tmp_4 = t_3;
		end
		tmp_3 = tmp_4;
	elseif (t_2 <= -2e-145)
		tmp_3 = t_2;
	elseif (t_2 <= 0.0)
		tmp_3 = t_0;
	elseif (t_2 <= 1e+240)
		tmp_3 = t_2;
	else
		tmp_3 = t_0;
	end
	return tmp_3
end

MATLAB

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

↓

function tmp_6 = code(a, b, c)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - b) / (a * 2.0);
	else
		tmp = -1.0 * (c / b);
	end
	t_0 = tmp;
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp_2 = 0.0;
	if (b >= 0.0)
		tmp_2 = (-b - t_1) / (2.0 * a);
	else
		tmp_2 = (2.0 * c) / (-b + t_1);
	end
	t_2 = tmp_2;
	t_3 = -1.0 * (b / a);
	tmp_4 = 0.0;
	if (t_2 <= -Inf)
		tmp_5 = 0.0;
		if (b >= 0.0)
			tmp_5 = t_3;
		else
			tmp_5 = t_3;
		end
		tmp_4 = tmp_5;
	elseif (t_2 <= -2e-145)
		tmp_4 = t_2;
	elseif (t_2 <= 0.0)
		tmp_4 = t_0;
	elseif (t_2 <= 1e+240)
		tmp_4 = t_2;
	else
		tmp_4 = t_0;
	end
	tmp_6 = tmp_4;
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], 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], 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]]

↓

code[a_, b_, c_] := Block[{t$95$0 = If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$1), $MachinePrecision]), $MachinePrecision]]}, Block[{t$95$3 = N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], If[GreaterEqual[b, 0.0], t$95$3, t$95$3], If[LessEqual[t$95$2, -2e-145], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$0, If[LessEqual[t$95$2, 1e+240], t$95$2, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Taylor expanded in b around inf 17.3

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

   4. Taylor expanded in c around 0 17.3

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

   5. Taylor expanded in b around 0 17.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]

   if -inf.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -1.99999999999999983e-145 or 0.0 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 1.00000000000000001e240

   1. Initial program 3.1

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

   if -1.99999999999999983e-145 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 0.0 or 1.00000000000000001e240 < (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))

   1. Initial program 36.8

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

   2. Simplified 36.8

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

      [Start] 36.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 36.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 36.8	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Taylor expanded in b around inf 29.4

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

   4. Taylor expanded in b around -inf 14.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \leq -2 \cdot 10^{-145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \leq 10^{+240}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	7756

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \frac{c \cdot 2}{\left(-b\right) + -1 \cdot b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+145}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-50}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.7
Cost	7696

\[\begin{array}{l} t_0 := \frac{c \cdot 2}{\left(-b\right) + -1 \cdot b}\\ t_1 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-36}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(c \cdot a\right) \cdot -4} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} + -1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	18.1
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-36}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c + c}{\sqrt{\left(c \cdot a\right) \cdot -4} - b}\\ \end{array} \]

Alternative 4
Error	22.7
Cost	644

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

Alternative 5
Error	45.0
Cost	452

\[\begin{array}{l} t_0 := -1 \cdot \frac{b}{a}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Reproduce

herbie shell --seed 2023090 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))