Average Error: 33.3 → 9.9
Time: 4.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.8813430075089506 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.35146897748971308 \cdot 10^{79}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) + \left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.8813430075089506 \cdot 10^{-61}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.35146897748971308 \cdot 10^{79}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) + \left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r61246 = b_2;
        double r61247 = -r61246;
        double r61248 = r61246 * r61246;
        double r61249 = a;
        double r61250 = c;
        double r61251 = r61249 * r61250;
        double r61252 = r61248 - r61251;
        double r61253 = sqrt(r61252);
        double r61254 = r61247 - r61253;
        double r61255 = r61254 / r61249;
        return r61255;
}
↓
double f(double a, double b_2, double c) {
        double r61256 = b_2;
        double r61257 = -2.8813430075089506e-61;
        bool r61258 = r61256 <= r61257;
        double r61259 = -0.5;
        double r61260 = c;
        double r61261 = r61260 / r61256;
        double r61262 = r61259 * r61261;
        double r61263 = 7.351468977489713e+79;
        bool r61264 = r61256 <= r61263;
        double r61265 = a;
        double r61266 = r61256 / r61265;
        double r61267 = -r61266;
        double r61268 = r61256 * r61256;
        double r61269 = r61265 * r61260;
        double r61270 = r61268 - r61269;
        double r61271 = sqrt(r61270);
        double r61272 = r61271 / r61265;
        double r61273 = -r61272;
        double r61274 = r61267 + r61273;
        double r61275 = 0.5;
        double r61276 = r61275 * r61261;
        double r61277 = 2.0;
        double r61278 = r61277 * r61266;
        double r61279 = r61276 - r61278;
        double r61280 = r61264 ? r61274 : r61279;
        double r61281 = r61258 ? r61262 : r61280;
        return r61281;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.8813430075089506e-61

    1. Initial program 53.2

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

    2. Taylor expanded around -inf 8.4

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

    if -2.8813430075089506e-61 < b_2 < 7.351468977489713e+79

    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. Using strategy rm

    5. Applied div-inv13.5

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

    6. Applied add-cube-cbrt13.5

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

    7. Applied times-frac13.5

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

    8. Simplified13.5

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

    9. Simplified13.5

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

    11. Applied sub-neg13.5

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

    12. Applied distribute-lft-in13.5

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

    13. Simplified13.4

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

    14. Simplified13.3

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

    if 7.351468977489713e+79 < b_2

    1. Initial program 41.8

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

    2. Taylor expanded around inf 4.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.8813430075089506 \cdot 10^{-61}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.35146897748971308 \cdot 10^{79}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) + \left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))