Average Error: 34.0 → 7.0
Time: 5.7s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.889385504355394 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.015156662498609 \cdot 10^{-179}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 1.3393452413162394 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{-c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -4.889385504355394 \cdot 10^{+152}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq -2.015156662498609 \cdot 10^{-179}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \leq 1.3393452413162394 \cdot 10^{+49}:\\
\;\;\;\;\frac{1}{\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{-c}}\\

\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 -4.889385504355394e+152)
   (- (* 0.5 (/ c b_2)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 -2.015156662498609e-179)
     (* (- (sqrt (- (* b_2 b_2) (* c a))) b_2) (/ 1.0 a))
     (if (<= b_2 1.3393452413162394e+49)
       (/ 1.0 (* (+ b_2 (sqrt (- (* b_2 b_2) (* c a)))) (/ 1.0 (- c))))
       (* (/ 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 <= -4.889385504355394e+152) {
		tmp = (0.5 * (c / b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= -2.015156662498609e-179) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) - b_2) * (1.0 / a);
	} else if (b_2 <= 1.3393452413162394e+49) {
		tmp = 1.0 / ((b_2 + sqrt((b_2 * b_2) - (c * a))) * (1.0 / -c));
	} 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 4 regimes

  2. if b_2 < -4.8893855043553942e152

    1. Initial program 63.4

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

    2. Simplified63.4

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

    3. Taylor expanded around -inf 2.9

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

    if -4.8893855043553942e152 < b_2 < -2.01515666249860915e-179

    1. Initial program 6.0

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

    2. Simplified6.0

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

    4. Applied div-inv_binary646.2

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

    if -2.01515666249860915e-179 < b_2 < 1.3393452413162394e49

    1. Initial program 26.1

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

    2. Simplified26.1

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

    4. Applied flip--_binary6426.3

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

    5. Simplified16.6

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

    6. Simplified16.6

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

    8. Applied clear-num_binary6416.7

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

    9. Simplified11.3

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

    if 1.3393452413162394e49 < b_2

    1. Initial program 57.2

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

    2. Simplified57.2

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

    3. Taylor expanded around inf 4.1

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.889385504355394 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -2.015156662498609 \cdot 10^{-179}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 1.3393452413162394 \cdot 10^{+49}:\\ \;\;\;\;\frac{1}{\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{-c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020299 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))