Average Error: 34.4 → 10.2
Time: 7.1s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.648624456952728 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.828439544463213 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -7.648624456952728 \cdot 10^{+146}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 1.828439544463213 \cdot 10^{-130}:\\
\;\;\;\;\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 -7.648624456952728e+146)
   (- (* 0.5 (/ c b_2)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 1.828439544463213e-130)
     (/ (- (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 <= -7.648624456952728e+146) {
		tmp = (0.5 * (c / b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 1.828439544463213e-130) {
		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 < -7.6486244569527281e146

    1. Initial program 61.1

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

    2. Simplified61.1

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

    3. Taylor expanded around -inf 2.8

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

    if -7.6486244569527281e146 < b_2 < 1.82843954446321311e-130

    1. Initial program 10.6

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

    2. Simplified10.6

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

    4. Applied *-un-lft-identity_binary6410.6

      \[\leadsto \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{\color{blue}{1 \cdot a}}\]

    5. Applied *-un-lft-identity_binary6410.6

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{1 \cdot a}\]

    6. Applied times-frac_binary6410.6

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    if 1.82843954446321311e-130 < b_2

    1. Initial program 51.8

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

    2. Simplified51.8

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

    3. Taylor expanded around inf 11.9

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.648624456952728 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.828439544463213 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))