Average Error: 34.9 → 7.7
Time: 6.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.598568989113781 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.135816095786639 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.1764143448038173 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{a} \cdot \left(a \cdot \left(-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.598568989113781 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 5.135816095786639 \cdot 10^{-282}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 4.1764143448038173 \cdot 10^{+80}:\\
\;\;\;\;\frac{1}{a} \cdot \left(a \cdot \left(-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)\right)\\

\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 -2.598568989113781e+152)
   (- (/ c b) (/ b a))
   (if (<= b 5.135816095786639e-282)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (if (<= b 4.1764143448038173e+80)
       (*
        (/ 1.0 a)
        (* a (* -2.0 (/ c (+ b (sqrt (- (* b b) (* c (* a 4.0)))))))))
       (- (/ 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 <= -2.598568989113781e+152) {
		tmp = (c / b) - (b / a);
	} else if (b <= 5.135816095786639e-282) {
		tmp = (sqrt((b * b) - (c * (a * 4.0))) - b) / (a * 2.0);
	} else if (b <= 4.1764143448038173e+80) {
		tmp = (1.0 / a) * (a * (-2.0 * (c / (b + sqrt((b * b) - (c * (a * 4.0)))))));
	} 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 < -2.5985689891137809e152

    1. Initial program 63.4

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

    2. Simplified63.4

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

    3. Taylor expanded around -inf 1.9

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

    if -2.5985689891137809e152 < b < 5.1358160957866387e-282

    1. Initial program 8.7

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

    2. Simplified8.7

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

    if 5.1358160957866387e-282 < b < 4.1764143448038173e80

    1. Initial program 33.2

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

    2. Simplified33.2

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

    4. Applied flip--_binary64_38933.2

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

    5. Simplified16.0

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

    6. Simplified16.0

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

    8. Applied *-un-lft-identity_binary64_41416.0

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

    9. Applied times-frac_binary64_42016.1

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

    10. Simplified13.6

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

    if 4.1764143448038173e80 < b

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around inf 2.8

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

    4. Simplified2.8

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.598568989113781 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 5.135816095786639 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 4.1764143448038173 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{a} \cdot \left(a \cdot \left(-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))