Average Error: 34.2 → 11.8
Time: 18.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r20204 = b_2;
        double r20205 = -r20204;
        double r20206 = r20204 * r20204;
        double r20207 = a;
        double r20208 = c;
        double r20209 = r20207 * r20208;
        double r20210 = r20206 - r20209;
        double r20211 = sqrt(r20210);
        double r20212 = r20205 + r20211;
        double r20213 = r20212 / r20207;
        return r20213;
}
↓
double f(double a, double b_2, double c) {
        double r20214 = b_2;
        double r20215 = -2.007820467288354e+70;
        bool r20216 = r20214 <= r20215;
        double r20217 = c;
        double r20218 = r20217 / r20214;
        double r20219 = 0.5;
        double r20220 = a;
        double r20221 = r20214 / r20220;
        double r20222 = -2.0;
        double r20223 = r20221 * r20222;
        double r20224 = fma(r20218, r20219, r20223);
        double r20225 = 7.455592343308264e-170;
        bool r20226 = r20214 <= r20225;
        double r20227 = 1.0;
        double r20228 = r20227 / r20220;
        double r20229 = r20214 * r20214;
        double r20230 = r20220 * r20217;
        double r20231 = r20229 - r20230;
        double r20232 = sqrt(r20231);
        double r20233 = r20232 - r20214;
        double r20234 = r20228 * r20233;
        double r20235 = -0.5;
        double r20236 = r20235 * r20218;
        double r20237 = r20226 ? r20234 : r20236;
        double r20238 = r20216 ? r20224 : r20237;
        return r20238;
}

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 < -2.007820467288354e+70

    1. Initial program 41.4

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

    2. Simplified41.4

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

    3. Taylor expanded around -inf 4.9

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

    4. Simplified4.9

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

    if -2.007820467288354e+70 < b_2 < 7.455592343308264e-170

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied clear-num12.1

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

    6. Applied div-inv12.2

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

    7. Applied add-cube-cbrt12.2

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

    8. Applied times-frac12.1

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

    9. Simplified12.1

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

    10. Simplified12.1

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

    if 7.455592343308264e-170 < b_2

    1. Initial program 48.9

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

    2. Simplified48.9

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

    3. Taylor expanded around inf 14.1

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

  4. Final simplification11.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))