Average Error: 34.0 → 9.8
Time: 12.1s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4123968497632831 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \mathbf{elif}\;b_2 \leq 1.7698566509785747 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.4123968497632831 \cdot 10^{+153}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\

\mathbf{elif}\;b_2 \leq 1.7698566509785747 \cdot 10^{-104}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot -0.5\\

\end{array}
(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.4123968497632831e+153)
   (+ (* 0.5 (/ c b_2)) (* (/ b_2 a) -2.0))
   (if (<= b_2 1.7698566509785747e-104)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* (/ c b_2) -0.5))))
double code(double a, double b_2, double c) {
	return (-b_2 + sqrt((b_2 * b_2) - (a * c))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4123968497632831e+153) {
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0);
	} else if (b_2 <= 1.7698566509785747e-104) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

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_2 < -1.4123968497632831e153

    1. Initial program 63.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified63.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 2.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    4. Simplified2.1

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2}\]

    if -1.4123968497632831e153 < b_2 < 1.76985665097857468e-104

    1. Initial program 11.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified11.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    if 1.76985665097857468e-104 < b_2

    1. Initial program 52.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified52.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 10.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4123968497632831 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \mathbf{elif}\;b_2 \leq 1.7698566509785747 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))