Average Error: 34.5 → 9.4
Time: 7.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -4.270668456863539 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9440742229641566 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 2.2542282560766223 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -4.270668456863539 \cdot 10^{+83}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 3.9440742229641566 \cdot 10^{-203}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\

\mathbf{elif}\;b \leq 2.2542282560766223 \cdot 10^{+78}:\\
\;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\

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

\end{array}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -4.270668456863539e+83)
   (- (/ c b) (/ b a))
   (if (<= b 3.9440742229641566e-203)
     (/ 1.0 (/ a (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) 2.0)))
     (if (<= b 2.2542282560766223e+78)
       (/
        (/ (* (* c a) -4.0) (+ b (sqrt (- (* b b) (* c (* a 4.0))))))
        (* a 2.0))
       (- (/ c b))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.270668456863539e+83) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.9440742229641566e-203) {
		tmp = 1.0 / (a / ((sqrt((b * b) - (c * (a * 4.0))) - b) / 2.0));
	} else if (b <= 2.2542282560766223e+78) {
		tmp = (((c * a) * -4.0) / (b + sqrt((b * b) - (c * (a * 4.0))))) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

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 4 regimes

  2. if b < -4.27066845686353916e83

    1. Initial program 43.5

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

    2. Simplified43.5

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

    3. Taylor expanded around -inf 4.8

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -4.27066845686353916e83 < b < 3.9440742229641566e-203

    1. Initial program 11.3

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

    2. Simplified11.3

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

    4. Applied clear-num_binary6411.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}\]

    5. Simplified11.4

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}}}\]

    if 3.9440742229641566e-203 < b < 2.25422825607662234e78

    1. Initial program 37.0

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

    2. Simplified37.0

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

    4. Applied flip--_binary6437.0

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

    5. Simplified17.0

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

    6. Simplified17.0

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

    if 2.25422825607662234e78 < b

    1. Initial program 58.1

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

    2. Simplified58.1

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

    3. Taylor expanded around inf 3.2

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    4. Simplified3.2

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.270668456863539 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9440742229641566 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 2.2542282560766223 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020253 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))