Average Error: 34.2 → 10.4
Time: 13.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\

\mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r16485 = b_2;
        double r16486 = -r16485;
        double r16487 = r16485 * r16485;
        double r16488 = a;
        double r16489 = c;
        double r16490 = r16488 * r16489;
        double r16491 = r16487 - r16490;
        double r16492 = sqrt(r16491);
        double r16493 = r16486 + r16492;
        double r16494 = r16493 / r16488;
        return r16494;
}
↓
double f(double a, double b_2, double c) {
        double r16495 = b_2;
        double r16496 = -4.123103533644211e+95;
        bool r16497 = r16495 <= r16496;
        double r16498 = 0.5;
        double r16499 = c;
        double r16500 = r16499 / r16495;
        double r16501 = -2.0;
        double r16502 = r16495 * r16501;
        double r16503 = a;
        double r16504 = r16502 / r16503;
        double r16505 = fma(r16498, r16500, r16504);
        double r16506 = 3.446447862996811e-75;
        bool r16507 = r16495 <= r16506;
        double r16508 = 1.0;
        double r16509 = cbrt(r16508);
        double r16510 = r16509 * r16509;
        double r16511 = r16495 * r16495;
        double r16512 = r16503 * r16499;
        double r16513 = r16511 - r16512;
        double r16514 = sqrt(r16513);
        double r16515 = r16514 - r16495;
        double r16516 = r16503 / r16515;
        double r16517 = r16510 / r16516;
        double r16518 = -0.5;
        double r16519 = r16518 * r16500;
        double r16520 = r16507 ? r16517 : r16519;
        double r16521 = r16497 ? r16505 : r16520;
        return r16521;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -4.123103533644211e+95

    1. Initial program 47.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    3. Simplified3.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)}\]

    if -4.123103533644211e+95 < b_2 < 3.446447862996811e-75

    1. Initial program 13.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num13.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    4. Simplified13.4

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt13.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    7. Applied associate-/l*13.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\sqrt[3]{1}}}}\]

    8. Simplified13.4

      \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\color{blue}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

    if 3.446447862996811e-75 < b_2

    1. Initial program 52.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 9.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b_2 \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))