Average Error: 34.5 → 10.0
Time: 7.7s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.2934209015436434 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.478299413084075 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -6.2934209015436434 \cdot 10^{+131}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 2.478299413084075 \cdot 10^{-134}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{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 -6.2934209015436434e+131)
   (+ (* 0.5 (/ c b_2)) (* -2.0 (/ b_2 a)))
   (if (<= b_2 2.478299413084075e-134)
     (- (/ (sqrt (- (* b_2 b_2) (* c a))) 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 <= -6.2934209015436434e+131) {
		tmp = (0.5 * (c / b_2)) + (-2.0 * (b_2 / a));
	} else if (b_2 <= 2.478299413084075e-134) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) / 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 < -6.293420901543643e131

    1. Initial program 56.8

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

    2. Simplified56.8

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

    3. Taylor expanded around -inf 2.6

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

    4. Simplified2.6

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

    if -6.293420901543643e131 < b_2 < 2.47829941308407479e-134

    1. Initial program 10.4

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

    2. Simplified10.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
    3. Using strategy rm

    4. Applied div-sub_binary6410.4

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

    5. Simplified10.4

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

    if 2.47829941308407479e-134 < b_2

    1. Initial program 51.3

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

    2. Simplified51.3

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

    3. Taylor expanded around inf 11.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.2934209015436434 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + -2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.478299413084075 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))