Average Error: 33.7 → 6.5
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.27507998502897628 \cdot 10^{133}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.0003811195914403 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 1.77072263135177874 \cdot 10^{102}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.27507998502897628 \cdot 10^{133}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.0003811195914403 \cdot 10^{-269}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\

\mathbf{elif}\;b_2 \le 1.77072263135177874 \cdot 10^{102}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
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 VAR;
	if ((b_2 <= -1.2750799850289763e+133)) {
		VAR = (-0.5 * (c / b_2));
	} else {
		double VAR_1;
		if ((b_2 <= 3.0003811195914403e-269)) {
			VAR_1 = ((1.0 / (sqrt(((b_2 * b_2) - (a * c))) - b_2)) / (1.0 / c));
		} else {
			double VAR_2;
			if ((b_2 <= 1.7707226313517787e+102)) {
				VAR_2 = ((-b_2 / a) - (sqrt(((b_2 * b_2) - (a * c))) / a));
			} else {
				VAR_2 = ((0.5 * (c / b_2)) - (2.0 * (b_2 / a)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -1.2750799850289763e+133

    1. Initial program 62.2

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

    2. Taylor expanded around -inf 1.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.2750799850289763e+133 < b_2 < 3.0003811195914403e-269

    1. Initial program 32.0

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

    3. Applied flip--32.0

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

    4. Simplified15.3

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c, a, 0\right)}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.3

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c, a, 0\right)}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied div-inv15.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, a, 0\right) \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    8. Using strategy rm

    9. Applied *-commutative15.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \mathsf{fma}\left(c, a, 0\right)}}{a}\]

    10. Applied associate-/l*14.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{a}{\mathsf{fma}\left(c, a, 0\right)}}}\]

    11. Taylor expanded around 0 8.2

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

    if 3.0003811195914403e-269 < b_2 < 1.7707226313517787e+102

    1. Initial program 9.0

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

    3. Applied div-sub9.0

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

    if 1.7707226313517787e+102 < b_2

    1. Initial program 47.4

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

    2. Taylor expanded around inf 3.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.27507998502897628 \cdot 10^{133}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.0003811195914403 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{1}{c}}\\ \mathbf{elif}\;b_2 \le 1.77072263135177874 \cdot 10^{102}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))