Average Error: 33.6 → 9.3
Time: 4.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.60254369165666142 \cdot 10^{59}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.4658143772560903 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\sqrt[3]{\frac{1}{a}}}}\\ \mathbf{elif}\;b_2 \le 3.1827258823065575 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.60254369165666142 \cdot 10^{59}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.4658143772560903 \cdot 10^{-105}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\sqrt[3]{\frac{1}{a}}}}\\

\mathbf{elif}\;b_2 \le 3.1827258823065575 \cdot 10^{65}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;-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 <= -1.6025436916566614e+59)) {
		VAR = (-0.5 * (c / b_2));
	} else {
		double VAR_1;
		if ((b_2 <= -6.46581437725609e-105)) {
			VAR_1 = ((cbrt((1.0 / a)) * cbrt((1.0 / a))) / ((1.0 / ((0.0 + (a * c)) / (sqrt(((b_2 * b_2) - (a * c))) - b_2))) / cbrt((1.0 / a))));
		} else {
			double VAR_2;
			if ((b_2 <= 3.1827258823065575e+65)) {
				VAR_2 = (1.0 / (a / (-b_2 - sqrt(((b_2 * b_2) - (a * c))))));
			} else {
				VAR_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 < -1.6025436916566614e+59

    1. Initial program 57.4

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

    2. Taylor expanded around -inf 3.8

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

    if -1.6025436916566614e+59 < b_2 < -6.46581437725609e-105

    1. Initial program 39.7

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

    3. Applied clear-num39.8

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

    5. Applied div-inv39.8

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

    6. Applied associate-/r*39.8

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

    8. Applied add-cube-cbrt40.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}\right) \cdot \sqrt[3]{\frac{1}{a}}}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    9. Applied associate-/l*40.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\sqrt[3]{\frac{1}{a}}}}}\]
    10. Using strategy rm

    11. Applied flip--40.1

      \[\leadsto \frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}{\sqrt[3]{\frac{1}{a}}}}\]

    12. Simplified15.8

      \[\leadsto \frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\sqrt[3]{\frac{1}{a}}}}\]

    13. Simplified15.8

      \[\leadsto \frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}{\sqrt[3]{\frac{1}{a}}}}\]

    if -6.46581437725609e-105 < b_2 < 3.1827258823065575e+65

    1. Initial program 12.4

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

    3. Applied clear-num12.6

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

    if 3.1827258823065575e+65 < b_2

    1. Initial program 39.6

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

    3. Applied clear-num39.7

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

    4. Taylor expanded around 0 5.3

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.60254369165666142 \cdot 10^{59}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.4658143772560903 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{1}{a}}}{\frac{\frac{1}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{\sqrt[3]{\frac{1}{a}}}}\\ \mathbf{elif}\;b_2 \le 3.1827258823065575 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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))