Average Error: 20.3 → 6.8
Time: 5.7s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.085320093966694026309795848133442393253 \cdot 10^{114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -6.724716266922556499072228000203116587273 \cdot 10^{-252}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.263391352577237531613239682082042403214 \cdot 10^{98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.085320093966694026309795848133442393253 \cdot 10^{114}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le -6.724716266922556499072228000203116587273 \cdot 10^{-252}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \le 5.263391352577237531613239682082042403214 \cdot 10^{98}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r44182 = b;
        double r44183 = 0.0;
        bool r44184 = r44182 >= r44183;
        double r44185 = 2.0;
        double r44186 = c;
        double r44187 = r44185 * r44186;
        double r44188 = -r44182;
        double r44189 = r44182 * r44182;
        double r44190 = 4.0;
        double r44191 = a;
        double r44192 = r44190 * r44191;
        double r44193 = r44192 * r44186;
        double r44194 = r44189 - r44193;
        double r44195 = sqrt(r44194);
        double r44196 = r44188 - r44195;
        double r44197 = r44187 / r44196;
        double r44198 = r44188 + r44195;
        double r44199 = r44185 * r44191;
        double r44200 = r44198 / r44199;
        double r44201 = r44184 ? r44197 : r44200;
        return r44201;
}
↓
double f(double a, double b, double c) {
        double r44202 = b;
        double r44203 = -1.085320093966694e+114;
        bool r44204 = r44202 <= r44203;
        double r44205 = 0.0;
        bool r44206 = r44202 >= r44205;
        double r44207 = 2.0;
        double r44208 = c;
        double r44209 = r44207 * r44208;
        double r44210 = -r44202;
        double r44211 = r44202 * r44202;
        double r44212 = 4.0;
        double r44213 = a;
        double r44214 = r44212 * r44213;
        double r44215 = r44214 * r44208;
        double r44216 = r44211 - r44215;
        double r44217 = sqrt(r44216);
        double r44218 = r44210 - r44217;
        double r44219 = r44209 / r44218;
        double r44220 = cbrt(r44202);
        double r44221 = r44220 * r44220;
        double r44222 = r44213 / r44221;
        double r44223 = r44208 / r44220;
        double r44224 = r44222 * r44223;
        double r44225 = r44207 * r44224;
        double r44226 = 2.0;
        double r44227 = r44226 * r44202;
        double r44228 = r44225 - r44227;
        double r44229 = r44207 * r44213;
        double r44230 = r44228 / r44229;
        double r44231 = r44206 ? r44219 : r44230;
        double r44232 = -6.7247162669225565e-252;
        bool r44233 = r44202 <= r44232;
        double r44234 = 1.0;
        double r44235 = r44210 + r44210;
        double r44236 = r44234 + r44235;
        double r44237 = -r44234;
        double r44238 = r44237 / r44234;
        double r44239 = r44236 + r44238;
        double r44240 = r44209 / r44239;
        double r44241 = r44210 + r44217;
        double r44242 = r44241 / r44229;
        double r44243 = r44206 ? r44240 : r44242;
        double r44244 = 5.2633913525772375e+98;
        bool r44245 = r44202 <= r44244;
        double r44246 = -r44216;
        double r44247 = fma(r44202, r44202, r44246);
        double r44248 = r44247 / r44218;
        double r44249 = r44248 / r44229;
        double r44250 = r44206 ? r44219 : r44249;
        double r44251 = r44245 ? r44250 : r44243;
        double r44252 = r44233 ? r44243 : r44251;
        double r44253 = r44204 ? r44231 : r44252;
        return r44253;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.085320093966694e+114

    1. Initial program 50.6

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Taylor expanded around -inf 11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt11.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{a \cdot c}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}} - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    5. Applied times-frac4.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}{2 \cdot a}\\ \end{array}\]

    if -1.085320093966694e+114 < b < -6.7247162669225565e-252 or 5.2633913525772375e+98 < b

    1. Initial program 19.2

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied expm1-log1p-u19.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    4. Taylor expanded around inf 7.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{1 - \left(b + \left(e^{-\log \left(\frac{1}{b}\right)} + \frac{e^{-\log \left(\frac{1}{b}\right)}}{b}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    5. Simplified6.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    if -6.7247162669225565e-252 < b < 5.2633913525772375e+98

    1. Initial program 9.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
    2. Using strategy rm

    3. Applied flip-+9.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]

    4. Simplified9.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.085320093966694026309795848133442393253 \cdot 10^{114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \frac{c}{\sqrt[3]{b}}\right) - 2 \cdot b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -6.724716266922556499072228000203116587273 \cdot 10^{-252}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 5.263391352577237531613239682082042403214 \cdot 10^{98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(1 + \left(\left(-b\right) + \left(-b\right)\right)\right) + \frac{-1}{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))