Average Error: 34.0 → 7.0
Time: 4.1s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9668071883221095 \cdot 10^{+93}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.470994545983059 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.3162613383615781 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot 0.5 - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.9668071883221095 \cdot 10^{+93}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq -4.470994545983059 \cdot 10^{-292}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\

\mathbf{elif}\;b_2 \leq 1.3162613383615781 \cdot 10^{+131}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot 0.5 - 2 \cdot \frac{b_2}{a}\\

\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 -1.9668071883221095e+93)
   (* -0.5 (/ c b_2))
   (if (<= b_2 -4.470994545983059e-292)
     (/ 1.0 (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) c))
     (if (<= b_2 1.3162613383615781e+131)
       (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
       (- (* (/ c b_2) 0.5) (* 2.0 (/ b_2 a)))))))
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 <= -1.9668071883221095e+93) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.470994545983059e-292) {
		tmp = 1.0 / ((sqrt((b_2 * b_2) - (c * a)) - b_2) / c);
	} else if (b_2 <= 1.3162613383615781e+131) {
		tmp = (-b_2 / a) - (sqrt((b_2 * b_2) - (c * a)) / a);
	} else {
		tmp = ((c / b_2) * 0.5) - (2.0 * (b_2 / a));
	}
	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 < -1.9668071883221095e93

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.0

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

    if -1.9668071883221095e93 < b_2 < -4.470994545983059e-292

    1. Initial program 32.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--_binary6432.4

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied clear-num_binary6416.3

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

    8. Simplified8.9

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

    if -4.470994545983059e-292 < b_2 < 1.3162613383615781e131

    1. Initial program 9.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub_binary649.4

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

    if 1.3162613383615781e131 < b_2

    1. Initial program 56.1

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

    2. Taylor expanded around inf 3.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.9668071883221095 \cdot 10^{+93}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -4.470994545983059 \cdot 10^{-292}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}\\ \mathbf{elif}\;b_2 \leq 1.3162613383615781 \cdot 10^{+131}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot 0.5 - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))