Average Error: 33.7 → 9.7
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.8621678288459276 \cdot 10^{80}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.07965843072761894 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.8621678288459276 \cdot 10^{80}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -5.07965843072761894 \cdot 10^{-161}:\\
\;\;\;\;\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1.2919983862558445 \cdot 10^{30}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\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 <= -4.862167828845928e+80)) {
		VAR = (-0.5 * (c / b_2));
	} else {
		double VAR_1;
		if ((b_2 <= -5.079658430727619e-161)) {
			VAR_1 = (((0.0 + (a * c)) / (sqrt(((b_2 * b_2) - (a * c))) - b_2)) / a);
		} else {
			double VAR_2;
			if ((b_2 <= 1.2919983862558445e+30)) {
				VAR_2 = (1.0 / (a / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))));
			} 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 < -4.862167828845928e+80

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 3.1

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

    if -4.862167828845928e+80 < b_2 < -5.079658430727619e-161

    1. Initial program 36.9

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

    3. Applied add-exp-log40.7

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a}\]
    4. Using strategy rm

    5. Applied flip--40.7

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

    6. Simplified18.7

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + e^{\log \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a}\]

    7. Simplified16.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -5.079658430727619e-161 < b_2 < 1.2919983862558445e+30

    1. Initial program 12.6

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

    3. Applied clear-num12.7

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

    if 1.2919983862558445e+30 < b_2

    1. Initial program 34.2

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

    2. Taylor expanded around inf 6.6

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.8621678288459276 \cdot 10^{80}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.07965843072761894 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.2919983862558445 \cdot 10^{30}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))