Average Error: 34.0 → 8.9
Time: 6.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.625373603204043 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.6856742787563955 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 7.957206871783919 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.625373603204043 \cdot 10^{+153}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 2.6856742787563955 \cdot 10^{-139}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\

\mathbf{elif}\;b \leq 7.957206871783919 \cdot 10^{+17}:\\
\;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{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 -3.625373603204043e+153)
   (- (/ c b) (/ b a))
   (if (<= b 2.6856742787563955e-139)
     (/ 1.0 (/ a (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) 2.0)))
     (if (<= b 7.957206871783919e+17)
       (/
        (/ (* a (* c -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 <= -3.625373603204043e+153) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.6856742787563955e-139) {
		tmp = 1.0 / (a / ((sqrt((b * b) - (c * (a * 4.0))) - b) / 2.0));
	} else if (b <= 7.957206871783919e+17) {
		tmp = ((a * (c * -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 < -3.6253736032040433e153

    1. Initial program 63.6

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

    2. Simplified63.6

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

    3. Taylor expanded around -inf 2.5

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

    if -3.6253736032040433e153 < b < 2.68567427875639553e-139

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied clear-num_binary64_41311.2

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

    5. Simplified11.2

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

    if 2.68567427875639553e-139 < b < 795720687178391936

    1. Initial program 34.5

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

    2. Simplified34.5

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

    4. Applied flip--_binary64_38934.5

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

    5. Simplified15.6

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

    6. Simplified15.6

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

    if 795720687178391936 < b

    1. Initial program 56.5

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

    2. Simplified56.5

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

    3. Taylor expanded around inf 5.3

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

    4. Simplified5.3

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.625373603204043 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.6856742787563955 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2}}}\\ \mathbf{elif}\;b \leq 7.957206871783919 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))