Average Error: 34.2 → 11.8
Time: 17.7s
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 r20166 = b_2;
        double r20167 = -r20166;
        double r20168 = r20166 * r20166;
        double r20169 = a;
        double r20170 = c;
        double r20171 = r20169 * r20170;
        double r20172 = r20168 - r20171;
        double r20173 = sqrt(r20172);
        double r20174 = r20167 + r20173;
        double r20175 = r20174 / r20169;
        return r20175;
}
↓
double f(double a, double b_2, double c) {
        double r20176 = b_2;
        double r20177 = -2.007820467288354e+70;
        bool r20178 = r20176 <= r20177;
        double r20179 = c;
        double r20180 = r20179 / r20176;
        double r20181 = 0.5;
        double r20182 = a;
        double r20183 = r20176 / r20182;
        double r20184 = -2.0;
        double r20185 = r20183 * r20184;
        double r20186 = fma(r20180, r20181, r20185);
        double r20187 = 7.455592343308264e-170;
        bool r20188 = r20176 <= r20187;
        double r20189 = 1.0;
        double r20190 = r20189 / r20182;
        double r20191 = r20176 * r20176;
        double r20192 = r20182 * r20179;
        double r20193 = r20191 - r20192;
        double r20194 = sqrt(r20193);
        double r20195 = r20194 - r20176;
        double r20196 = r20190 * r20195;
        double r20197 = -0.5;
        double r20198 = r20197 * r20180;
        double r20199 = r20188 ? r20196 : r20198;
        double r20200 = r20178 ? r20186 : r20199;
        return r20200;
}

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))