Average Error: 33.1 → 10.3
Time: 23.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6124768939899423 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.6124768939899423 \cdot 10^{+64}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1175192 = b;
        double r1175193 = -r1175192;
        double r1175194 = r1175192 * r1175192;
        double r1175195 = 4.0;
        double r1175196 = a;
        double r1175197 = r1175195 * r1175196;
        double r1175198 = c;
        double r1175199 = r1175197 * r1175198;
        double r1175200 = r1175194 - r1175199;
        double r1175201 = sqrt(r1175200);
        double r1175202 = r1175193 + r1175201;
        double r1175203 = 2.0;
        double r1175204 = r1175203 * r1175196;
        double r1175205 = r1175202 / r1175204;
        return r1175205;
}
↓
double f(double a, double b, double c) {
        double r1175206 = b;
        double r1175207 = -1.6124768939899423e+64;
        bool r1175208 = r1175206 <= r1175207;
        double r1175209 = c;
        double r1175210 = r1175209 / r1175206;
        double r1175211 = a;
        double r1175212 = r1175206 / r1175211;
        double r1175213 = r1175210 - r1175212;
        double r1175214 = 2.0;
        double r1175215 = r1175213 * r1175214;
        double r1175216 = r1175215 / r1175214;
        double r1175217 = 9.831724396970673e-110;
        bool r1175218 = r1175206 <= r1175217;
        double r1175219 = 1.0;
        double r1175220 = r1175219 / r1175211;
        double r1175221 = -4.0;
        double r1175222 = r1175211 * r1175221;
        double r1175223 = r1175222 * r1175209;
        double r1175224 = fma(r1175206, r1175206, r1175223);
        double r1175225 = sqrt(r1175224);
        double r1175226 = r1175225 - r1175206;
        double r1175227 = r1175220 * r1175226;
        double r1175228 = r1175227 / r1175214;
        double r1175229 = -2.0;
        double r1175230 = r1175229 * r1175210;
        double r1175231 = r1175230 / r1175214;
        double r1175232 = r1175218 ? r1175228 : r1175231;
        double r1175233 = r1175208 ? r1175216 : r1175232;
        return r1175233;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6124768939899423e+64

    1. Initial program 37.7

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

    2. Simplified37.7

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

    3. Taylor expanded around -inf 5.2

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

    4. Simplified5.2

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

    if -1.6124768939899423e+64 < b < 9.831724396970673e-110

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-inv12.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{1}{a}}}{2}\]

    if 9.831724396970673e-110 < b

    1. Initial program 51.0

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

    2. Simplified51.0

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

    3. Taylor expanded around inf 10.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6124768939899423 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))