Average Error: 34.0 → 14.0
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.427668844436332 \cdot 10^{79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5 \cdot \frac{a}{b}, c, -2 \cdot b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.62273985298184147 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.427668844436332 \cdot 10^{79}:\\
\;\;\;\;\frac{\mathsf{fma}\left(1.5 \cdot \frac{a}{b}, c, -2 \cdot b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le 3.62273985298184147 \cdot 10^{-79}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
}
↓
double code(double a, double b, double c) {
	double temp;
	if ((b <= -2.4276688444363324e+79)) {
		temp = (fma((1.5 * (a / b)), c, (-2.0 * b)) / (3.0 * a));
	} else {
		double temp_1;
		if ((b <= 3.6227398529818415e-79)) {
			temp_1 = (fma((cbrt(b) * cbrt(b)), -cbrt(b), sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
		} else {
			temp_1 = (-(1.5 * ((a * c) / b)) / (3.0 * a));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.4276688444363324e+79

    1. Initial program 43.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt44.1

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

    4. Applied distribute-rgt-neg-in44.1

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

    5. Applied fma-def44.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

    6. Taylor expanded around -inf 11.5

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

    7. Simplified5.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5 \cdot \frac{a}{b}, c, -2 \cdot b\right)}}{3 \cdot a}\]

    if -2.4276688444363324e+79 < b < 3.6227398529818415e-79

    1. Initial program 12.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt12.4

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

    4. Applied distribute-rgt-neg-in12.4

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

    5. Applied fma-def12.4

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

    if 3.6227398529818415e-79 < b

    1. Initial program 53.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt56.4

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

    4. Applied distribute-rgt-neg-in56.4

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

    5. Applied fma-def57.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

    6. Taylor expanded around inf 19.4

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.427668844436332 \cdot 10^{79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5 \cdot \frac{a}{b}, c, -2 \cdot b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 3.62273985298184147 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{b} \cdot \sqrt[3]{b}, -\sqrt[3]{b}, \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))