Average Error: 19.1 → 12.6
Time: 2.0m
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.0416169482605103 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right) + \mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt[3]{a}}{\sqrt{b}} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b}}{2}}{a}\\ \end{array}\]
\begin{array}{l}
\mathbf{if}\;b \ge 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 3.0416169482605103 \cdot 10^{+99}:\\
\;\;\;\;\begin{array}{l}
\mathbf{if}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right) + \mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right)}\\

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

\end{array}\\

\mathbf{elif}\;b \ge 0:\\
\;\;\;\;\frac{2 \cdot c}{\left(\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt[3]{a}}{\sqrt{b}} - b\right) \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1384308 = b;
        double r1384309 = 0.0;
        bool r1384310 = r1384308 >= r1384309;
        double r1384311 = 2.0;
        double r1384312 = c;
        double r1384313 = r1384311 * r1384312;
        double r1384314 = -r1384308;
        double r1384315 = r1384308 * r1384308;
        double r1384316 = 4.0;
        double r1384317 = a;
        double r1384318 = r1384316 * r1384317;
        double r1384319 = r1384318 * r1384312;
        double r1384320 = r1384315 - r1384319;
        double r1384321 = sqrt(r1384320);
        double r1384322 = r1384314 - r1384321;
        double r1384323 = r1384313 / r1384322;
        double r1384324 = r1384314 + r1384321;
        double r1384325 = r1384311 * r1384317;
        double r1384326 = r1384324 / r1384325;
        double r1384327 = r1384310 ? r1384323 : r1384326;
        return r1384327;
}
↓
double f(double a, double b, double c) {
        double r1384328 = b;
        double r1384329 = 3.0416169482605103e+99;
        bool r1384330 = r1384328 <= r1384329;
        double r1384331 = 0.0;
        bool r1384332 = r1384328 >= r1384331;
        double r1384333 = 2.0;
        double r1384334 = c;
        double r1384335 = r1384333 * r1384334;
        double r1384336 = -4.0;
        double r1384337 = a;
        double r1384338 = r1384337 * r1384334;
        double r1384339 = r1384328 * r1384328;
        double r1384340 = fma(r1384336, r1384338, r1384339);
        double r1384341 = sqrt(r1384340);
        double r1384342 = sqrt(r1384341);
        double r1384343 = -r1384342;
        double r1384344 = r1384342 * r1384342;
        double r1384345 = fma(r1384343, r1384342, r1384344);
        double r1384346 = sqrt(r1384328);
        double r1384347 = -r1384346;
        double r1384348 = r1384342 * r1384343;
        double r1384349 = fma(r1384346, r1384347, r1384348);
        double r1384350 = r1384345 + r1384349;
        double r1384351 = r1384335 / r1384350;
        double r1384352 = r1384341 - r1384328;
        double r1384353 = r1384352 / r1384333;
        double r1384354 = r1384353 / r1384337;
        double r1384355 = r1384332 ? r1384351 : r1384354;
        double r1384356 = cbrt(r1384337);
        double r1384357 = r1384356 * r1384356;
        double r1384358 = r1384357 / r1384346;
        double r1384359 = r1384334 * r1384358;
        double r1384360 = r1384356 / r1384346;
        double r1384361 = r1384359 * r1384360;
        double r1384362 = r1384361 - r1384328;
        double r1384363 = r1384362 * r1384333;
        double r1384364 = r1384335 / r1384363;
        double r1384365 = r1384341 * r1384341;
        double r1384366 = r1384365 - r1384339;
        double r1384367 = r1384341 + r1384328;
        double r1384368 = r1384366 / r1384367;
        double r1384369 = r1384368 / r1384333;
        double r1384370 = r1384369 / r1384337;
        double r1384371 = r1384332 ? r1384364 : r1384370;
        double r1384372 = r1384330 ? r1384355 : r1384371;
        return r1384372;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 3.0416169482605103e+99

    1. Initial program 15.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified15.7

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt15.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    5. Applied add-sqr-sqrt15.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) - \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    6. Applied distribute-rgt-neg-in15.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\sqrt{b} \cdot \left(-\sqrt{b}\right)} - \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    7. Applied prod-diff15.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\right) + \mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    if 3.0416169482605103e+99 < b

    1. Initial program 29.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Simplified29.6

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]

    3. Taylor expanded around inf 5.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    4. Simplified2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \left(c \cdot \frac{a}{b} - b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(c \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    7. Applied add-cube-cbrt2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(c \cdot \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\sqrt{b} \cdot \sqrt{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    8. Applied times-frac2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(c \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}} \cdot \frac{\sqrt[3]{a}}{\sqrt{b}}\right)} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

    9. Applied associate-*r*2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt[3]{a}}{\sqrt{b}}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
    10. Using strategy rm

    11. Applied flip--2.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt[3]{a}}{\sqrt{b}} - b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2}}{a}\\ \end{array}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.0416169482605103 \cdot 10^{+99}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right) + \mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}} \cdot \left(-\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(\left(c \cdot \frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt[3]{a}}{\sqrt{b}} - b\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} + b}}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))