jeff quadratic root 1

Average Error: 19.5 → 6.6

Time: 20.6s

Precision: binary64

Cost: 7820

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ t_1 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.95 \cdot 10^{+86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

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 (/ (- (- b) b) (* a 2.0)))
        (t_1 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (<= b -1e+134)
     (if (>= b 0.0) t_0 (/ (* 2.0 c) (fma 2.0 (* a (/ c b)) (* b -2.0))))
     (if (<= b 7.95e+86)
       (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) (/ (* 2.0 c) (- t_1 b)))
       (if (>= b 0.0) t_0 (/ b a))))))

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 t_0 = (-b - b) / (a * 2.0);
	double t_1 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp_1;
	if (b <= -1e+134) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = t_0;
		} else {
			tmp_2 = (2.0 * c) / fma(2.0, (a * (c / b)), (b * -2.0));
		}
		tmp_1 = tmp_2;
	} else if (b <= 7.95e+86) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = (-b - t_1) / (a * 2.0);
		} else {
			tmp_3 = (2.0 * c) / (t_1 - b);
		}
		tmp_1 = tmp_3;
	} else if (b >= 0.0) {
		tmp_1 = t_0;
	} else {
		tmp_1 = b / a;
	}
	return tmp_1;
}

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)
	t_0 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	t_1 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -1e+134)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = t_0;
		else
			tmp_2 = Float64(Float64(2.0 * c) / fma(2.0, Float64(a * Float64(c / b)), Float64(b * -2.0)));
		end
		tmp_1 = tmp_2;
	elseif (b <= 7.95e+86)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
		else
			tmp_3 = Float64(Float64(2.0 * c) / Float64(t_1 - b));
		end
		tmp_1 = tmp_3;
	elseif (b >= 0.0)
		tmp_1 = t_0;
	else
		tmp_1 = Float64(b / a);
	end
	return tmp_1
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 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1e+134], If[GreaterEqual[b, 0.0], t$95$0, N[(N[(2.0 * c), $MachinePrecision] / N[(2.0 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 7.95e+86], If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], t$95$0, N[(b / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999921e133

   1. Initial program 33.6

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

   2. Simplified 33.6

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 c)) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (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) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 33.6

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

   4. Taylor expanded in b around -inf 6.4

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

   5. Simplified 1.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 b -2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b -2)): 25 points increase in error, 21 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 1.5

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

   if -9.99999999999999921e133 < b < 7.9499999999999996e86

   1. Initial program 8.7

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

   2. Simplified 8.7

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 c)) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (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) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error

   if 7.9499999999999996e86 < b

   1. Initial program 44.4

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

   2. Simplified 44.4

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))) (*.f64 a 2)) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= *-commutative_binary64 (*.f64 2 a))) (/.f64 (*.f64 c 2) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 c)) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 4 (*.f64 a c))))))): 0 points increase in error, 0 points decrease in error
      (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) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 4.4

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

   4. Taylor expanded in b around -inf 4.4

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

   5. Simplified 4.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 b -2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b -2)): 25 points increase in error, 21 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in c around inf 4.4

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+134}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.95 \cdot 10^{+86}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	7624

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -2 \cdot 10^{+136}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

Alternative 2
Error	18.2
Cost	7496

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -2.25 \cdot 10^{-9}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 3
Error	18.3
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;b \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 4
Error	44.6
Cost	644

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

Alternative 5
Error	22.5
Cost	644

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

Reproduce

herbie shell --seed 2022332 
(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)))))))