Average Error: 34.6 → 7.0
Time: 4.1s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.4839011960233026 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -7.399951737087807 \cdot 10^{-272}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 7.171533889314125 \cdot 10^{+60}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.4839011960233026 \cdot 10^{+51}:\\
\;\;\;\;\frac{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq -7.399951737087807 \cdot 10^{-272}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \leq 7.171533889314125 \cdot 10^{+60}:\\
\;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

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

\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 -2.4839011960233026e+51)
   (/ (- (- (* 0.5 (/ c (/ b_2 a))) b_2) b_2) a)
   (if (<= b_2 -7.399951737087807e-272)
     (* (- (sqrt (- (* b_2 b_2) (* c a))) b_2) (/ 1.0 a))
     (if (<= b_2 7.171533889314125e+60)
       (- (/ c (+ b_2 (sqrt (- (* b_2 b_2) (* c a))))))
       (* -0.5 (/ c b_2))))))
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 <= -2.4839011960233026e+51) {
		tmp = (((0.5 * (c / (b_2 / a))) - b_2) - b_2) / a;
	} else if (b_2 <= -7.399951737087807e-272) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) - b_2) * (1.0 / a);
	} else if (b_2 <= 7.171533889314125e+60) {
		tmp = -(c / (b_2 + sqrt((b_2 * b_2) - (c * a))));
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 4 regimes

  2. if b_2 < -2.4839011960233026e51

    1. Initial program 38.4

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

    2. Simplified38.4

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

    3. Taylor expanded around -inf 10.2

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \frac{a \cdot c}{b_2} - b_2\right)} - b_2}{a}\]

    4. Simplified5.2

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right)} - b_2}{a}\]

    if -2.4839011960233026e51 < b_2 < -7.39995173708780702e-272

    1. Initial program 8.5

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

    2. Simplified8.5

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

    4. Applied div-inv_binary648.6

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

    if -7.39995173708780702e-272 < b_2 < 7.17153388931412499e60

    1. Initial program 29.8

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

    2. Simplified29.8

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

    4. Applied flip--_binary6429.9

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

    5. Simplified18.0

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

    6. Simplified18.0

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

    8. Applied distribute-frac-neg_binary6418.0

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

    9. Applied distribute-frac-neg_binary6418.0

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

    10. Simplified10.2

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

    if 7.17153388931412499e60 < b_2

    1. Initial program 57.7

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 3.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.4839011960233026 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{c}{\frac{b_2}{a}} - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -7.399951737087807 \cdot 10^{-272}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 7.171533889314125 \cdot 10^{+60}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))