Average Error: 34.1 → 6.6
Time: 20.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\
\;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r84326 = b_2;
        double r84327 = -r84326;
        double r84328 = r84326 * r84326;
        double r84329 = a;
        double r84330 = c;
        double r84331 = r84329 * r84330;
        double r84332 = r84328 - r84331;
        double r84333 = sqrt(r84332);
        double r84334 = r84327 - r84333;
        double r84335 = r84334 / r84329;
        return r84335;
}
↓
double f(double a, double b_2, double c) {
        double r84336 = b_2;
        double r84337 = -3.5132588248780117e+152;
        bool r84338 = r84336 <= r84337;
        double r84339 = -0.5;
        double r84340 = c;
        double r84341 = r84340 / r84336;
        double r84342 = r84339 * r84341;
        double r84343 = 8.216265756828381e-276;
        bool r84344 = r84336 <= r84343;
        double r84345 = 2.0;
        double r84346 = pow(r84336, r84345);
        double r84347 = a;
        double r84348 = r84347 * r84340;
        double r84349 = r84346 - r84348;
        double r84350 = sqrt(r84349);
        double r84351 = r84350 - r84336;
        double r84352 = r84340 / r84351;
        double r84353 = 5.031608061939103e+53;
        bool r84354 = r84336 <= r84353;
        double r84355 = 1.0;
        double r84356 = -r84336;
        double r84357 = r84336 * r84336;
        double r84358 = r84357 - r84348;
        double r84359 = sqrt(r84358);
        double r84360 = r84356 - r84359;
        double r84361 = r84347 / r84360;
        double r84362 = r84355 / r84361;
        double r84363 = 0.5;
        double r84364 = r84336 / r84347;
        double r84365 = -2.0;
        double r84366 = r84364 * r84365;
        double r84367 = fma(r84363, r84341, r84366);
        double r84368 = r84354 ? r84362 : r84367;
        double r84369 = r84344 ? r84352 : r84368;
        double r84370 = r84338 ? r84342 : r84369;
        return r84370;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.5132588248780117e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.4

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

    if -3.5132588248780117e+152 < b_2 < 8.216265756828381e-276

    1. Initial program 33.1

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

    3. Applied flip--33.1

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

    4. Simplified15.2

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied div-inv15.3

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \frac{1}{a}}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity15.3

      \[\leadsto \color{blue}{\left(1 \cdot \frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\right)} \cdot \frac{1}{a}\]

    10. Applied associate-*l*15.3

      \[\leadsto \color{blue}{1 \cdot \left(\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \frac{1}{a}\right)}\]

    11. Simplified7.8

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

    if 8.216265756828381e-276 < b_2 < 5.031608061939103e+53

    1. Initial program 9.3

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

    3. Applied clear-num9.5

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

    if 5.031608061939103e+53 < b_2

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))