Average Error: 33.1 → 8.6
Time: 4.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.282303906287587 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\frac{a}{b} \cdot c\right) - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.9121987620533157 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 4.4076190914119784 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot \left(-c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -2.282303906287587 \cdot 10^{+153}:\\
\;\;\;\;\frac{\left(2 \cdot \left(\frac{a}{b} \cdot c\right) - b\right) - b}{2 \cdot a}\\

\mathbf{elif}\;b \leq 2.9121987620533157 \cdot 10^{-123}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\

\mathbf{elif}\;b \leq 4.4076190914119784 \cdot 10^{+69}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot \left(-c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double VAR;
	if ((b <= -2.282303906287587e+153)) {
		VAR = (((double) (((double) (((double) (2.0 * ((double) ((a / b) * c)))) - b)) - b)) / ((double) (2.0 * a)));
	} else {
		double VAR_1;
		if ((b <= 2.9121987620533157e-123)) {
			VAR_1 = (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))) - b)) / ((double) (2.0 * a)));
		} else {
			double VAR_2;
			if ((b <= 4.4076190914119784e+69)) {
				VAR_2 = ((((double) (4.0 * ((double) (a * ((double) -(c)))))) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c))))))))))) / ((double) (2.0 * a)));
			} else {
				VAR_2 = ((double) (-1.0 * (c / b)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.1
Target20.9
Herbie8.6
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.28230390628758713e153

    1. Initial program Error: 63.5 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. SimplifiedError: 63.5 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

    3. Taylor expanded around -inf Error: 11.6 bits

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{a \cdot c}{b} - b\right)} - b}{a \cdot 2}\]

    4. SimplifiedError: 2.4 bits

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \left(\frac{a}{b} \cdot c\right) - b\right)} - b}{a \cdot 2}\]

    if -2.28230390628758713e153 < b < 2.9121987620533157e-123

    1. Initial program Error: 10.9 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. SimplifiedError: 10.9 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

    if 2.9121987620533157e-123 < b < 4.4076190914119784e69

    1. Initial program Error: 38.4 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. SimplifiedError: 38.4 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrtError: 40.0 bits

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - b}{a \cdot 2}\]
    5. Using strategy rm

    6. Applied flip--Error: 40.0 bits

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) - b \cdot b}{\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} + b}}}{a \cdot 2}\]

    7. SimplifiedError: 15.1 bits

      \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot \left(-c\right)\right)}}{\left(\sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} + b}}{a \cdot 2}\]

    8. SimplifiedError: 14.6 bits

      \[\leadsto \frac{\frac{4 \cdot \left(a \cdot \left(-c\right)\right)}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2}\]

    if 4.4076190914119784e69 < b

    1. Initial program Error: 57.5 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. SimplifiedError: 57.5 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]

    3. Taylor expanded around inf Error: 3.5 bits

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 8.6 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.282303906287587 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\frac{a}{b} \cdot c\right) - b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.9121987620533157 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 4.4076190914119784 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot \left(-c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected #f

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))