Average Error: 33.5 → 10.5
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -227369802444031.66:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1057501 = b;
        double r1057502 = -r1057501;
        double r1057503 = r1057501 * r1057501;
        double r1057504 = 4.0;
        double r1057505 = a;
        double r1057506 = r1057504 * r1057505;
        double r1057507 = c;
        double r1057508 = r1057506 * r1057507;
        double r1057509 = r1057503 - r1057508;
        double r1057510 = sqrt(r1057509);
        double r1057511 = r1057502 + r1057510;
        double r1057512 = 2.0;
        double r1057513 = r1057512 * r1057505;
        double r1057514 = r1057511 / r1057513;
        return r1057514;
}
↓
double f(double a, double b, double c) {
        double r1057515 = b;
        double r1057516 = -227369802444031.66;
        bool r1057517 = r1057515 <= r1057516;
        double r1057518 = c;
        double r1057519 = r1057518 / r1057515;
        double r1057520 = a;
        double r1057521 = r1057515 / r1057520;
        double r1057522 = r1057519 - r1057521;
        double r1057523 = 2.0;
        double r1057524 = r1057522 * r1057523;
        double r1057525 = r1057524 / r1057523;
        double r1057526 = 2.0617732603635578e-61;
        bool r1057527 = r1057515 <= r1057526;
        double r1057528 = -4.0;
        double r1057529 = r1057528 * r1057518;
        double r1057530 = r1057515 * r1057515;
        double r1057531 = fma(r1057520, r1057529, r1057530);
        double r1057532 = sqrt(r1057531);
        double r1057533 = r1057532 - r1057515;
        double r1057534 = r1057533 / r1057520;
        double r1057535 = r1057534 / r1057523;
        double r1057536 = -2.0;
        double r1057537 = r1057536 * r1057519;
        double r1057538 = r1057537 / r1057523;
        double r1057539 = r1057527 ? r1057535 : r1057538;
        double r1057540 = r1057517 ? r1057525 : r1057539;
        return r1057540;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -227369802444031.66

    1. Initial program 32.9

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

    2. Simplified32.9

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

    3. Taylor expanded around 0 32.9

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

    4. Taylor expanded around -inf 6.8

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

    5. Simplified6.8

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

    if -227369802444031.66 < b < 2.0617732603635578e-61

    1. Initial program 14.9

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

    2. Simplified14.9

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

    3. Taylor expanded around 0 15.0

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

    5. Applied clear-num15.0

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

    7. Applied *-un-lft-identity15.0

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

    8. Applied *-un-lft-identity15.0

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

    9. Applied times-frac15.0

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

    10. Applied add-cube-cbrt15.0

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2}\]

    11. Applied times-frac15.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}}{2}\]

    12. Simplified15.0

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}}{2}\]

    13. Simplified15.0

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

    if 2.0617732603635578e-61 < b

    1. Initial program 52.8

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

    2. Simplified52.8

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

    3. Taylor expanded around 0 52.8

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

    4. Taylor expanded around inf 8.3

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))