jeff quadratic root 2

Average Error: 19.6 → 7.1

Time: 22.0s

Precision: binary64

Cost: 38052

Math

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

↓

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \frac{\left(-b\right) + t_1}{2 \cdot a}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+238}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (if (>= b 0.0) (/ (* 2.0 c) (- (- b) b)) (/ (- b) a)))
        (t_1 (sqrt (- (* b b) (* (* 4.0 a) c))))
        (t_2 (/ (+ (- b) t_1) (* 2.0 a)))
        (t_3 (if (>= b 0.0) (/ (* 2.0 c) (- (- b) t_1)) t_2)))
   (if (<= t_3 (- INFINITY))
     t_0
     (if (<= t_3 -5e-238)
       t_3
       (if (<= t_3 0.0)
         (if (>= b 0.0)
           (/ (* 2.0 c) (- (- b) (+ b (* (/ c b) (* -2.0 a)))))
           t_2)
         (if (<= t_3 5e+238) t_3 t_0))))))

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 tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = -b / a;
	}
	double t_0 = tmp;
	double t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	double t_2 = (-b + t_1) / (2.0 * a);
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_1);
	} else {
		tmp_1 = t_2;
	}
	double t_3 = tmp_1;
	double tmp_2;
	if (t_3 <= -((double) INFINITY)) {
		tmp_2 = t_0;
	} else if (t_3 <= -5e-238) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - (b + ((c / b) * (-2.0 * a))));
		} else {
			tmp_3 = t_2;
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= 5e+238) {
		tmp_2 = t_3;
	} else {
		tmp_2 = t_0;
	}
	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 tmp;
	if (b >= 0.0) {
		tmp = (2.0 * c) / (-b - b);
	} else {
		tmp = -b / a;
	}
	double t_0 = tmp;
	double t_1 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double t_2 = (-b + t_1) / (2.0 * a);
	double tmp_1;
	if (b >= 0.0) {
		tmp_1 = (2.0 * c) / (-b - t_1);
	} else {
		tmp_1 = t_2;
	}
	double t_3 = tmp_1;
	double tmp_2;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp_2 = t_0;
	} else if (t_3 <= -5e-238) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (2.0 * c) / (-b - (b + ((c / b) * (-2.0 * a))));
		} else {
			tmp_3 = t_2;
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= 5e+238) {
		tmp_2 = t_3;
	} else {
		tmp_2 = t_0;
	}
	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):
	tmp = 0
	if b >= 0.0:
		tmp = (2.0 * c) / (-b - b)
	else:
		tmp = -b / a
	t_0 = tmp
	t_1 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	t_2 = (-b + t_1) / (2.0 * a)
	tmp_1 = 0
	if b >= 0.0:
		tmp_1 = (2.0 * c) / (-b - t_1)
	else:
		tmp_1 = t_2
	t_3 = tmp_1
	tmp_2 = 0
	if t_3 <= -math.inf:
		tmp_2 = t_0
	elif t_3 <= -5e-238:
		tmp_2 = t_3
	elif t_3 <= 0.0:
		tmp_3 = 0
		if b >= 0.0:
			tmp_3 = (2.0 * c) / (-b - (b + ((c / b) * (-2.0 * a))))
		else:
			tmp_3 = t_2
		tmp_2 = tmp_3
	elif t_3 <= 5e+238:
		tmp_2 = t_3
	else:
		tmp_2 = t_0
	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)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) - b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	t_0 = tmp
	t_1 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	t_2 = Float64(Float64(Float64(-b) + t_1) / Float64(2.0 * a))
	tmp_1 = 0.0
	if (b >= 0.0)
		tmp_1 = Float64(Float64(2.0 * c) / Float64(Float64(-b) - t_1));
	else
		tmp_1 = t_2;
	end
	t_3 = tmp_1
	tmp_2 = 0.0
	if (t_3 <= Float64(-Inf))
		tmp_2 = t_0;
	elseif (t_3 <= -5e-238)
		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(-b) - Float64(b + Float64(Float64(c / b) * Float64(-2.0 * a)))));
		else
			tmp_3 = t_2;
		end
		tmp_2 = tmp_3;
	elseif (t_3 <= 5e+238)
		tmp_2 = t_3;
	else
		tmp_2 = t_0;
	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)
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (2.0 * c) / (-b - b);
	else
		tmp = -b / a;
	end
	t_0 = tmp;
	t_1 = sqrt(((b * b) - ((4.0 * a) * c)));
	t_2 = (-b + t_1) / (2.0 * a);
	tmp_2 = 0.0;
	if (b >= 0.0)
		tmp_2 = (2.0 * c) / (-b - t_1);
	else
		tmp_2 = t_2;
	end
	t_3 = tmp_2;
	tmp_3 = 0.0;
	if (t_3 <= -Inf)
		tmp_3 = t_0;
	elseif (t_3 <= -5e-238)
		tmp_3 = t_3;
	elseif (t_3 <= 0.0)
		tmp_4 = 0.0;
		if (b >= 0.0)
			tmp_4 = (2.0 * c) / (-b - (b + ((c / b) * (-2.0 * a))));
		else
			tmp_4 = t_2;
		end
		tmp_3 = tmp_4;
	elseif (t_3 <= 5e+238)
		tmp_3 = t_3;
	else
		tmp_3 = t_0;
	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 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $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 = N[(N[((-b) + t$95$1), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]}, If[LessEqual[t$95$3, (-Infinity)], t$95$0, If[LessEqual[t$95$3, -5e-238], t$95$3, If[LessEqual[t$95$3, 0.0], If[GreaterEqual[b, 0.0], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) - N[(b + N[(N[(c / b), $MachinePrecision] * N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2], If[LessEqual[t$95$3, 5e+238], t$95$3, t$95$0]]]]]]]]

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 4.99999999999999995e238 < (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 57.5

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

   2. Taylor expanded in b around inf 55.6

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

   3. Taylor expanded in b around -inf 15.9

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

   4. Simplified 15.9

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

      [Start] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array} \]
      rational.json-simplify-2 [=>] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a} \cdot -1\\ \end{array} \]
      rational.json-simplify-9 [=>] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]
      rational.json-simplify-10 [=>] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1}\\ \end{array} \]
      rational.json-simplify-44 [=>] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{-1}}{a}\\ \end{array} \]
      rational.json-simplify-10 [<=] 15.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

   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-238 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))) < 4.99999999999999995e238

   1. Initial program 3.0

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

   if -5e-238 < (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 34.1

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

   2. Taylor expanded in b around inf 13.3

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

   3. Simplified 11.0

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

      [Start] 13.3	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \frac{c \cdot a}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
      rational.json-simplify-49 [=>] 10.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + -2 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
      rational.json-simplify-43 [=>] 10.9	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \color{blue}{a \cdot \left(\frac{c}{b} \cdot -2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
      rational.json-simplify-43 [=>] 11.0	\[ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \color{blue}{\frac{c}{b} \cdot \left(-2 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.1

   \[\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 - \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} \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 - \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} \leq -5 \cdot 10^{-238}:\\ \;\;\;\;\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}\\ \mathbf{elif}\;\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} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{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 - \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} \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\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}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t_1}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-270}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{c \cdot \left(2 \cdot \frac{a}{b}\right) + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+238}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.6
Cost	7888

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -9.4 \cdot 10^{-115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.3
Cost	7820

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+89}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{+55}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + t_0}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} + \left(-b\right)}{2 \cdot a}\\ \end{array} \]

Alternative 4
Error	13.9
Cost	7696

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \left(b + \frac{c}{b} \cdot \left(-2 \cdot a\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(-b\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;b \leq -9.4 \cdot 10^{-115}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(-b\right) + \frac{c}{b} \cdot \left(2 \cdot a\right)\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.8
Cost	7504

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq -3 \cdot 10^{-162}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{\sqrt{c \cdot \left(a \cdot -4\right)}} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(-b\right) + \frac{c}{b} \cdot \left(2 \cdot a\right)\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{c \cdot \left(2 \cdot \frac{a}{b}\right) + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

Alternative 6
Error	16.8
Cost	7504

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ t_1 := \frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq -2.45 \cdot 10^{-163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-2}{\frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{c \cdot \left(2 \cdot \frac{a}{b}\right) + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	16.7
Cost	7504

\[\begin{array}{l} t_0 := \frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{-163}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-110}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{c \cdot \left(a \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \left(\left(-b\right) + \frac{c}{b} \cdot \left(2 \cdot a\right)\right)}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{c \cdot \left(2 \cdot \frac{a}{b}\right) + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array} \]

Alternative 8
Error	23.1
Cost	7244

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{c \cdot \left(2 \cdot \frac{a}{b}\right) + b \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + -1 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{if}\;a \leq 4.6 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+89}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	22.6
Cost	1092

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

Alternative 10
Error	22.6
Cost	1092

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

Alternative 11
Error	22.6
Cost	1028

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

Alternative 12
Error	22.8
Cost	644

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

Alternative 13
Error	22.8
Cost	644

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

Reproduce

herbie shell --seed 2023075 
(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))))