Average Error: 19.9 → 13.0
Time: 18.3s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.070196910432930661212410040231827918751 \cdot 10^{84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}\right|}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)} - b}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
↓
\begin{array}{l}
\mathbf{if}\;b \le 2.070196910432930661212410040231827918751 \cdot 10^{84}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}\right|}{a \cdot 2}\\

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

\end{array}\\

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

\mathbf{else}:\\
\;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)} - b}\\

\end{array}
double f(double a, double b, double c) {
        double r35284 = b;
        double r35285 = 0.0;
        bool r35286 = r35284 >= r35285;
        double r35287 = -r35284;
        double r35288 = r35284 * r35284;
        double r35289 = 4.0;
        double r35290 = a;
        double r35291 = r35289 * r35290;
        double r35292 = c;
        double r35293 = r35291 * r35292;
        double r35294 = r35288 - r35293;
        double r35295 = sqrt(r35294);
        double r35296 = r35287 - r35295;
        double r35297 = 2.0;
        double r35298 = r35297 * r35290;
        double r35299 = r35296 / r35298;
        double r35300 = r35297 * r35292;
        double r35301 = r35287 + r35295;
        double r35302 = r35300 / r35301;
        double r35303 = r35286 ? r35299 : r35302;
        return r35303;
}
↓
double f(double a, double b, double c) {
        double r35304 = b;
        double r35305 = 2.0701969104329307e+84;
        bool r35306 = r35304 <= r35305;
        double r35307 = 0.0;
        bool r35308 = r35304 >= r35307;
        double r35309 = -r35304;
        double r35310 = a;
        double r35311 = 4.0;
        double r35312 = r35310 * r35311;
        double r35313 = c;
        double r35314 = -r35313;
        double r35315 = r35304 * r35304;
        double r35316 = fma(r35312, r35314, r35315);
        double r35317 = cbrt(r35316);
        double r35318 = sqrt(r35317);
        double r35319 = fabs(r35317);
        double r35320 = r35318 * r35319;
        double r35321 = r35309 - r35320;
        double r35322 = 2.0;
        double r35323 = r35310 * r35322;
        double r35324 = r35321 / r35323;
        double r35325 = r35313 * r35322;
        double r35326 = sqrt(r35316);
        double r35327 = r35326 - r35304;
        double r35328 = r35325 / r35327;
        double r35329 = r35308 ? r35324 : r35328;
        double r35330 = r35309 - r35304;
        double r35331 = r35330 / r35323;
        double r35332 = 1.0;
        double r35333 = fma(r35314, r35312, r35315);
        double r35334 = sqrt(r35333);
        double r35335 = r35334 - r35304;
        double r35336 = r35332 / r35335;
        double r35337 = r35325 * r35336;
        double r35338 = r35308 ? r35331 : r35337;
        double r35339 = r35306 ? r35329 : r35338;
        return r35339;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.0701969104329307e+84

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied add-cube-cbrt15.0

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

    5. Applied sqrt-prod15.0

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

    6. Simplified15.0

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

    if 2.0701969104329307e+84 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around 0 3.7

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

    5. Applied div-inv3.7

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

    6. Simplified3.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\left(2 \cdot c\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)} - b}}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.070196910432930661212410040231827918751 \cdot 10^{84}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)}\right|}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(a \cdot 4, -c, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot 2\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))