jeff quadratic root 1

Average Error: 20.0 → 7.2

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.252365262196174 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.276046310772245 \cdot 10^{-283}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.084166787237329 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(a \cdot c\right) \cdot 16\right)\right)}{b \cdot b + c \cdot \left(a \cdot 4\right)}} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{\sqrt[3]{b}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\]

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
 (if (<= b -6.252365262196174e+104)
   (if (>= b 0.0)
     (*
      -0.5
      (/ (+ b (- b (* 2.0 (* (/ a (* (cbrt b) (cbrt b))) (/ c (cbrt b)))))) a))
     (/ (* 2.0 c) (* 2.0 (- (* c (/ a b)) b))))
   (if (<= b -2.276046310772245e-283)
     (if (>= b 0.0)
       (*
        -0.5
        (*
         (/ (sqrt (* 2.0 (- b (/ (* a c) b)))) (* (cbrt a) (cbrt a)))
         (/ (sqrt (* 2.0 (- b (/ (* a c) b)))) (cbrt a))))
       (/ (* 2.0 c) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))
     (if (<= b 4.084166787237329e+98)
       (if (>= b 0.0)
         (* -0.5 (/ (+ b (sqrt (- (* b b) (* c (* a 4.0))))) a))
         (/
          (* 2.0 c)
          (-
           (sqrt
            (/
             (- (pow b 4.0) (* a (* c (* (* a c) 16.0))))
             (+ (* b b) (* c (* a 4.0)))))
           b)))
       (if (>= b 0.0)
         (*
          -0.5
          (/
           (+
            b
            (-
             b
             (*
              2.0
              (*
               (/ (/ a (* (cbrt b) (cbrt b))) (cbrt (* (cbrt b) (cbrt b))))
               (/ c (cbrt (cbrt b)))))))
           a))
         (/ (* 2.0 c) (- (sqrt (- (* b b) (* c (* a 4.0)))) 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 tmp;
	if (b <= -6.252365262196174e+104) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = -0.5 * ((b + (b - (2.0 * ((a / (cbrt(b) * cbrt(b))) * (c / cbrt(b)))))) / a);
		} else {
			tmp_1 = (2.0 * c) / (2.0 * ((c * (a / b)) - b));
		}
		tmp = tmp_1;
	} else if (b <= -2.276046310772245e-283) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = -0.5 * ((sqrt(2.0 * (b - ((a * c) / b))) / (cbrt(a) * cbrt(a))) * (sqrt(2.0 * (b - ((a * c) / b))) / cbrt(a)));
		} else {
			tmp_2 = (2.0 * c) / (sqrt((b * b) - (c * (a * 4.0))) - b);
		}
		tmp = tmp_2;
	} else if (b <= 4.084166787237329e+98) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b + sqrt((b * b) - (c * (a * 4.0)))) / a);
		} else {
			tmp_3 = (2.0 * c) / (sqrt((pow(b, 4.0) - (a * (c * ((a * c) * 16.0)))) / ((b * b) + (c * (a * 4.0)))) - b);
		}
		tmp = tmp_3;
	} else if (b >= 0.0) {
		tmp = -0.5 * ((b + (b - (2.0 * (((a / (cbrt(b) * cbrt(b))) / cbrt(cbrt(b) * cbrt(b))) * (c / cbrt(cbrt(b))))))) / a);
	} else {
		tmp = (2.0 * c) / (sqrt((b * b) - (c * (a * 4.0))) - b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -6.25236526219617385e104

   1. Initial program 31.0

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

   2. Simplified 31.0

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

   3. Taylor expanded around inf 31.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt_binary64_109 31.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a \cdot c}{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   6. Applied times-frac_binary64_83 31.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \color{blue}{\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   7. Taylor expanded around -inf 6.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

   8. Simplified 2.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}}\\ \end{array}\]

   if -6.25236526219617385e104 < b < -2.27604631077224517e-283

   1. Initial program 8.2

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

   2. Simplified 8.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

   3. Taylor expanded around inf 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt_binary64_109 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   6. Applied add-sqr-sqrt_binary64_98 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{\sqrt{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)} \cdot \sqrt{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   7. Applied times-frac_binary64_83 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \color{blue}{\left(\frac{\sqrt{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   8. Simplified 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\color{blue}{\frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\sqrt{b + \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   9. Simplified 8.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   if -2.27604631077224517e-283 < b < 4.0841667872373289e98

   1. Initial program 9.9

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

   2. Simplified 9.9

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied flip--_binary64_52 10.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{c} \cdot 2}{\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

   5. Simplified 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(a \cdot c\right) \cdot 16\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}} - b}\\ \end{array}\]

   if 4.0841667872373289e98 < b

   1. Initial program 46.9

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

   2. Simplified 46.9

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}}\]

   3. Taylor expanded around inf 10.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt_binary64_109 10.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \frac{a \cdot c}{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   6. Applied times-frac_binary64_83 4.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \color{blue}{\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt_binary64_109 4.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   9. Applied cbrt-prod_binary64_105 4.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\color{blue}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   10. Applied *-un-lft-identity_binary64_77 4.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{\color{blue}{1 \cdot c}}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\sqrt[3]{b}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   11. Applied times-frac_binary64_83 4.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{\sqrt[3]{b}}}\right)}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   12. Applied associate-*r*_binary64_19 4.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \color{blue}{\left(\left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{1}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \frac{c}{\sqrt[3]{\sqrt[3]{b}}}\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   13. Simplified 4.5

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\color{blue}{\frac{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}}}} \cdot \frac{c}{\sqrt[3]{\sqrt[3]{b}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
3. Recombined 4 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.252365262196174 \cdot 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.276046310772245 \cdot 10^{-283}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.084166787237329 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(a \cdot c\right) \cdot 16\right)\right)}{b \cdot b + c \cdot \left(a \cdot 4\right)}} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \left(b - 2 \cdot \left(\frac{\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}}}{\sqrt[3]{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{c}{\sqrt[3]{\sqrt[3]{b}}}\right)\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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)))))))