Average Error: 34.4 → 10.3
Time: 22.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1539564 = b;
        double r1539565 = -r1539564;
        double r1539566 = r1539564 * r1539564;
        double r1539567 = 4.0;
        double r1539568 = a;
        double r1539569 = r1539567 * r1539568;
        double r1539570 = c;
        double r1539571 = r1539569 * r1539570;
        double r1539572 = r1539566 - r1539571;
        double r1539573 = sqrt(r1539572);
        double r1539574 = r1539565 + r1539573;
        double r1539575 = 2.0;
        double r1539576 = r1539575 * r1539568;
        double r1539577 = r1539574 / r1539576;
        return r1539577;
}
↓
double f(double a, double b, double c) {
        double r1539578 = b;
        double r1539579 = -1.7633154797394035e+89;
        bool r1539580 = r1539578 <= r1539579;
        double r1539581 = a;
        double r1539582 = r1539578 / r1539581;
        double r1539583 = -2.0;
        double r1539584 = 2.0;
        double r1539585 = c;
        double r1539586 = r1539585 / r1539578;
        double r1539587 = r1539584 * r1539586;
        double r1539588 = fma(r1539582, r1539583, r1539587);
        double r1539589 = r1539588 / r1539584;
        double r1539590 = 9.136492990928292e-23;
        bool r1539591 = r1539578 <= r1539590;
        double r1539592 = 1.0;
        double r1539593 = r1539592 / r1539581;
        double r1539594 = r1539578 * r1539578;
        double r1539595 = 4.0;
        double r1539596 = r1539595 * r1539585;
        double r1539597 = r1539581 * r1539596;
        double r1539598 = r1539594 - r1539597;
        double r1539599 = sqrt(r1539598);
        double r1539600 = r1539599 - r1539578;
        double r1539601 = r1539593 * r1539600;
        double r1539602 = r1539601 / r1539584;
        double r1539603 = -2.0;
        double r1539604 = r1539586 * r1539603;
        double r1539605 = r1539604 / r1539584;
        double r1539606 = r1539591 ? r1539602 : r1539605;
        double r1539607 = r1539580 ? r1539589 : r1539606;
        return r1539607;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around -inf 3.9

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

    4. Simplified3.9

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

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.0

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

    2. Simplified15.1

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

    4. Applied div-inv15.2

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.5

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 6.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))