jeff quadratic root 1

Average Error: 19.8 → 6.5

Time: 14.6s

Precision: binary64

Cost: 51364

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ t_1 := \sqrt{b \cdot b + t_0}\\ t_2 := \frac{c \cdot 2}{t_1 - b}\\ t_3 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ t_4 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{t_0}\right)}}\right)}^{2}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt[3]{{\left(\mathsf{fma}\left(b, b, t_0\right)\right)}^{1.5}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 (* c (* a -4.0)))
        (t_1 (sqrt (+ (* b b) t_0)))
        (t_2 (/ (* c 2.0) (- t_1 b)))
        (t_3 (if (>= b 0.0) (/ (- (- b) t_1) (* a 2.0)) t_2))
        (t_4 (/ (- (- b) b) (* a 2.0))))
   (if (<= t_3 (- INFINITY))
     (if (>= b 0.0)
       t_4
       (pow (sqrt (/ (* c -2.0) (- b (hypot b (sqrt t_0))))) 2.0))
     (if (<= t_3 -2e-300)
       t_3
       (if (<= t_3 0.0)
         (if (>= b 0.0)
           (/ (- (- b) (cbrt (pow (fma b b t_0) 1.5))) (* a 2.0))
           (/ (* c 2.0) (- (- (* (* a 2.0) (/ c b)) b) b)))
         (if (<= t_3 2e+226)
           (if (>= b 0.0)
             (/
              (-
               (- b)
               (sqrt
                (+
                 (* b b)
                 (fma c (* a -4.0) (fma c (* a -4.0) (* a (* 4.0 c)))))))
              (* a 2.0))
             t_2)
           (if (>= b 0.0) t_4 (/ (- c) b))))))))

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 = c * (a * -4.0);
	double t_1 = sqrt(((b * b) + t_0));
	double t_2 = (c * 2.0) / (t_1 - b);
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_1) / (a * 2.0);
	} else {
		tmp = t_2;
	}
	double t_3 = tmp;
	double t_4 = (-b - b) / (a * 2.0);
	double tmp_2;
	if (t_3 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_4;
		} else {
			tmp_3 = pow(sqrt(((c * -2.0) / (b - hypot(b, sqrt(t_0))))), 2.0);
		}
		tmp_2 = tmp_3;
	} else if (t_3 <= -2e-300) {
		tmp_2 = t_3;
	} else if (t_3 <= 0.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - cbrt(pow(fma(b, b, t_0), 1.5))) / (a * 2.0);
		} else {
			tmp_4 = (c * 2.0) / ((((a * 2.0) * (c / b)) - b) - b);
		}
		tmp_2 = tmp_4;
	} else if (t_3 <= 2e+226) {
		double tmp_5;
		if (b >= 0.0) {
			tmp_5 = (-b - sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), (a * (4.0 * c))))))) / (a * 2.0);
		} else {
			tmp_5 = t_2;
		}
		tmp_2 = tmp_5;
	} else if (b >= 0.0) {
		tmp_2 = t_4;
	} else {
		tmp_2 = -c / b;
	}
	return tmp_2;
}

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(c * Float64(a * -4.0))
	t_1 = sqrt(Float64(Float64(b * b) + t_0))
	t_2 = Float64(Float64(c * 2.0) / Float64(t_1 - b))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0));
	else
		tmp = t_2;
	end
	t_3 = tmp
	t_4 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_2 = 0.0
	if (t_3 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_4;
		else
			tmp_3 = sqrt(Float64(Float64(c * -2.0) / Float64(b - hypot(b, sqrt(t_0))))) ^ 2.0;
		end
		tmp_2 = tmp_3;
	elseif (t_3 <= -2e-300)
		tmp_2 = t_3;
	elseif (t_3 <= 0.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - cbrt((fma(b, b, t_0) ^ 1.5))) / Float64(a * 2.0));
		else
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(Float64(Float64(a * 2.0) * Float64(c / b)) - b) - b));
		end
		tmp_2 = tmp_4;
	elseif (t_3 <= 2e+226)
		tmp_5 = 0.0
		if (b >= 0.0)
			tmp_5 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), Float64(a * Float64(4.0 * c))))))) / Float64(a * 2.0));
		else
			tmp_5 = t_2;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_4;
	else
		tmp_2 = Float64(Float64(-c) / b);
	end
	return tmp_2
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[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]}, Block[{t$95$4 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], If[GreaterEqual[b, 0.0], t$95$4, N[Power[N[Sqrt[N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], If[LessEqual[t$95$3, -2e-300], t$95$3, If[LessEqual[t$95$3, 0.0], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Power[N[Power[N[(b * b + t$95$0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[(N[(a * 2.0), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$3, 2e+226], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], t$95$4, N[((-c) / b), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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. Taylor expanded in b around inf 17.4

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

   3. Applied egg-rr 17.4

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

   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)))))) < -2.00000000000000005e-300

   1. Initial program 2.1

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

   if -2.00000000000000005e-300 < (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

   1. Initial program 37.6

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

   2. Taylor expanded in b around -inf 11.5

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

   3. Simplified 9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c}{b} \cdot \left(a \cdot 2\right) - b\right)}}\\ \end{array} \]
      Proof
      (-.f64 (*.f64 (/.f64 c b) (*.f64 a 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 c b) a) 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 c a) b)) 2) b): 23 points increase in error, 13 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b))) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (neg.f64 b))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 9.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \left(\frac{c}{b} \cdot \left(a \cdot 2\right) - b\right)}\\ \end{array} \]

   if 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)))))) < 1.99999999999999992e226

   1. Initial program 2.7

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

   2. Applied egg-rr 2.8

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

   if 1.99999999999999992e226 < (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 50.8

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

   2. Taylor expanded in b around inf 17.7

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

   3. Taylor expanded in b around -inf 14.8

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

   4. Simplified 14.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 c)) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\right)}^{2}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt[3]{{\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)\right)}^{1.5}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(\left(a \cdot 2\right) \cdot \frac{c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022291 
(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)))))))