Average Error: 33.8 → 9.6
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6844644503075447 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.6844644503075447 \cdot 10^{+144}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r718217 = b;
        double r718218 = -r718217;
        double r718219 = r718217 * r718217;
        double r718220 = 4.0;
        double r718221 = a;
        double r718222 = r718220 * r718221;
        double r718223 = c;
        double r718224 = r718222 * r718223;
        double r718225 = r718219 - r718224;
        double r718226 = sqrt(r718225);
        double r718227 = r718218 + r718226;
        double r718228 = 2.0;
        double r718229 = r718228 * r718221;
        double r718230 = r718227 / r718229;
        return r718230;
}
↓
double f(double a, double b, double c) {
        double r718231 = b;
        double r718232 = -1.6844644503075447e+144;
        bool r718233 = r718231 <= r718232;
        double r718234 = c;
        double r718235 = r718234 / r718231;
        double r718236 = a;
        double r718237 = r718231 / r718236;
        double r718238 = r718235 - r718237;
        double r718239 = 2.0;
        double r718240 = r718238 * r718239;
        double r718241 = r718240 / r718239;
        double r718242 = 1.739098950628615e-79;
        bool r718243 = r718231 <= r718242;
        double r718244 = -4.0;
        double r718245 = r718236 * r718244;
        double r718246 = r718245 * r718234;
        double r718247 = fma(r718231, r718231, r718246);
        double r718248 = sqrt(r718247);
        double r718249 = r718248 / r718236;
        double r718250 = r718249 - r718237;
        double r718251 = r718250 / r718239;
        double r718252 = 1.8656332031849816e-25;
        bool r718253 = r718231 <= r718252;
        double r718254 = -2.0;
        double r718255 = r718254 * r718235;
        double r718256 = r718255 / r718239;
        double r718257 = 5.297236684235463e-16;
        bool r718258 = r718231 <= r718257;
        double r718259 = r718258 ? r718251 : r718256;
        double r718260 = r718253 ? r718256 : r718259;
        double r718261 = r718243 ? r718251 : r718260;
        double r718262 = r718233 ? r718241 : r718261;
        return r718262;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6844644503075447e+144

    1. Initial program 58.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified58.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around -inf 2.5

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

    4. Simplified2.5

      \[\leadsto \frac{\color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}}{2}\]

    if -1.6844644503075447e+144 < b < 1.739098950628615e-79 or 1.8656332031849816e-25 < b < 5.297236684235463e-16

    1. Initial program 12.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified12.3

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied div-sub12.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}}{2}\]

    if 1.739098950628615e-79 < b < 1.8656332031849816e-25 or 5.297236684235463e-16 < b

    1. Initial program 53.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified53.2

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around inf 8.2

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6844644503075447 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))