Average Error: 43.7 → 11.1
Time: 6.5s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 0.0001664603697344127:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq 0.0001664603697344127:\\
\;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\

\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) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double VAR;
	if ((b <= 0.0001664603697344127)) {
		VAR = ((((double) (((double) (b * b)) - ((double) (((double) (b * b)) + ((double) (4.0 * ((double) (a * c)))))))) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c))))))))))) / ((double) (a * 2.0)));
	} else {
		VAR = ((double) (-1.0 * (c / b)));
	}
	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

Derivation

  1. Split input into 2 regimes

  2. if b < 1.66460369734412712e-4

    1. Initial program 18.6

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

    2. Simplified18.6

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

    4. Applied flip--18.6

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

    5. Simplified17.7

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

    6. Simplified17.7

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

    if 1.66460369734412712e-4 < b

    1. Initial program 45.6

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

    2. Simplified45.6

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

    3. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0001664603697344127:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))