Average Error: 34.1 → 11.2
Time: 12.9s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.334085620187398 \cdot 10^{+39}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.111603012388429 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.334085620187398 \cdot 10^{+39}:\\
\;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\

\mathbf{elif}\;b \leq 2.111603012388429 \cdot 10^{-140}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2} - \frac{b}{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 -3.334085620187398e+39)
   (/ (* b -2.0) (* a 2.0))
   (if (<= b 2.111603012388429e-140)
     (- (/ (sqrt (- (* b b) (* (* a 4.0) c))) (* a 2.0)) (/ b (* 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 <= -3.334085620187398e+39) {
		tmp = (b * -2.0) / (a * 2.0);
	} else if (b <= 2.111603012388429e-140) {
		tmp = (sqrt((b * b) - ((a * 4.0) * c)) / (a * 2.0)) - (b / (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 3 regimes

  2. if b < -3.3340856201873979e39

    1. Initial program 36.4

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

    2. Simplified36.4

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

    3. Taylor expanded around -inf 6.2

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{a \cdot 2}\]

    if -3.3340856201873979e39 < b < 2.11160301238842913e-140

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-sub_binary64_42412.1

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

    if 2.11160301238842913e-140 < b

    1. Initial program 50.1

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

    2. Simplified50.1

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

    3. Taylor expanded around inf 12.8

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

    4. Simplified12.8

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.334085620187398 \cdot 10^{+39}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.111603012388429 \cdot 10^{-140}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))