jeff quadratic root 1

Average Error: 19.9 → 8.1

Time: 6.6s

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 -1.0288136512639993 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)} \cdot \sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.0127078141426725 \cdot 10^{-167}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -9.704665002374583 \cdot 10^{-213}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)} \cdot \sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.234913879083865 \cdot 10^{+67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\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 -1.0288136512639993e+92)
   (if (>= b 0.0)
     (*
      -0.5
      (*
       (/
        (*
         (cbrt (* 2.0 (- b (* (/ c b) a))))
         (cbrt (* 2.0 (- b (* (/ c b) a)))))
        (* (cbrt a) (cbrt a)))
       (/ (cbrt (* 2.0 (- b (* (/ c b) a)))) (cbrt a))))
     (/
      (* 2.0 c)
      (- (- (* 2.0 (* (/ a (* (cbrt b) (cbrt b))) (/ c (cbrt b)))) b) b)))
   (if (<= b -4.0127078141426725e-167)
     (if (>= b 0.0)
       (/ (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))) (* 2.0 a))
       (/ (* 2.0 c) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))
     (if (<= b -9.704665002374583e-213)
       (if (>= b 0.0)
         (*
          -0.5
          (*
           (/
            (*
             (cbrt (* 2.0 (- b (* (/ c b) a))))
             (cbrt (* 2.0 (- b (* (/ c b) a)))))
            (* (cbrt a) (cbrt a)))
           (/ (cbrt (* 2.0 (- b (* (/ c b) a)))) (cbrt a))))
         (/
          (* 2.0 c)
          (- (- (* 2.0 (* (/ a (* (cbrt b) (cbrt b))) (/ c (cbrt b)))) b) b)))
       (if (<= b 1.234913879083865e+67)
         (if (>= b 0.0)
           (/ (- (- b) (sqrt (- (* b b) (* c (* a 4.0))))) (* 2.0 a))
           (/ (* 2.0 c) (- (sqrt (- (* b b) (* c (* a 4.0)))) b)))
         (if (>= b 0.0)
           (* -0.5 (* 2.0 (- (/ b a) (/ c b))))
           (/
            (* 2.0 c)
            (-
             (- (* 2.0 (* (/ a (* (cbrt b) (cbrt b))) (/ c (cbrt b)))) b)
             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 <= -1.0288136512639993e+92) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = -0.5 * (((cbrt(2.0 * (b - ((c / b) * a))) * cbrt(2.0 * (b - ((c / b) * a)))) / (cbrt(a) * cbrt(a))) * (cbrt(2.0 * (b - ((c / b) * a))) / cbrt(a)));
		} else {
			tmp_1 = (2.0 * c) / (((2.0 * ((a / (cbrt(b) * cbrt(b))) * (c / cbrt(b)))) - b) - b);
		}
		tmp = tmp_1;
	} else if (b <= -4.0127078141426725e-167) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - sqrt((b * b) - (c * (a * 4.0)))) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (sqrt((b * b) - (c * (a * 4.0))) - b);
		}
		tmp = tmp_2;
	} else if (b <= -9.704665002374583e-213) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * (((cbrt(2.0 * (b - ((c / b) * a))) * cbrt(2.0 * (b - ((c / b) * a)))) / (cbrt(a) * cbrt(a))) * (cbrt(2.0 * (b - ((c / b) * a))) / cbrt(a)));
		} else {
			tmp_3 = (2.0 * c) / (((2.0 * ((a / (cbrt(b) * cbrt(b))) * (c / cbrt(b)))) - b) - b);
		}
		tmp = tmp_3;
	} else if (b <= 1.234913879083865e+67) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - sqrt((b * b) - (c * (a * 4.0)))) / (2.0 * a);
		} else {
			tmp_4 = (2.0 * c) / (sqrt((b * b) - (c * (a * 4.0))) - b);
		}
		tmp = tmp_4;
	} else if (b >= 0.0) {
		tmp = -0.5 * (2.0 * ((b / a) - (c / b)));
	} else {
		tmp = (2.0 * c) / (((2.0 * ((a / (cbrt(b) * cbrt(b))) * (c / cbrt(b)))) - b) - b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.0288136512639993e92 or -4.0127078141426725e-167 < b < -9.7046650023745833e-213

   1. Initial program 27.8

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

   2. Simplified 27.8

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

   3. Taylor expanded around -inf 11.9

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

   5. Applied add-cube-cbrt_binary64_113 11.9

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

   6. Applied times-frac_binary64_84 8.4

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

   7. Taylor expanded around inf 8.4

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

   8. Simplified 8.4

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

   10. Applied add-cube-cbrt_binary64_113 8.4

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

   11. Applied add-cube-cbrt_binary64_113 8.4

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

   12. Applied times-frac_binary64_84 8.4

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

   13. Simplified 8.4

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

   14. Simplified 8.4

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

   if -1.0288136512639993e92 < b < -4.0127078141426725e-167 or -9.7046650023745833e-213 < b < 1.234913879083865e67

   1. Initial program 8.9

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

   if 1.234913879083865e67 < b

   1. Initial program 40.7

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

   2. Simplified 40.7

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

   3. Taylor expanded around -inf 40.7

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

   5. Applied add-cube-cbrt_binary64_113 40.7

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

   6. Applied times-frac_binary64_84 40.7

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

   7. Taylor expanded around inf 9.9

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

   8. Simplified 5.1

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

   10. Applied *-un-lft-identity_binary64_78 5.1

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

   11. Applied *-un-lft-identity_binary64_78 5.1

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

   12. Applied times-frac_binary64_84 5.1

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

   13. Simplified 5.1

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

   14. Simplified 5.0

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.0288136512639993 \cdot 10^{+92}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)} \cdot \sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -4.0127078141426725 \cdot 10^{-167}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -9.704665002374583 \cdot 10^{-213}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(\frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)} \cdot \sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{2 \cdot \left(b - \frac{c}{b} \cdot a\right)}}{\sqrt[3]{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.234913879083865 \cdot 10^{+67}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - b\right) - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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)))))))