Average Error: 34.1 → 10.2
Time: 37.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.2287897258283204 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \mathbf{elif}\;b_2 \le 1.4954103314709843 \cdot 10^{+136}:\\ \;\;\;\;-\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.2287897258283204 \cdot 10^{-38}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\

\mathbf{elif}\;b_2 \le 1.4954103314709843 \cdot 10^{+136}:\\
\;\;\;\;-\frac{c}{\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 r1176288 = b_2;
        double r1176289 = -r1176288;
        double r1176290 = r1176288 * r1176288;
        double r1176291 = a;
        double r1176292 = c;
        double r1176293 = r1176291 * r1176292;
        double r1176294 = r1176290 - r1176293;
        double r1176295 = sqrt(r1176294);
        double r1176296 = r1176289 + r1176295;
        double r1176297 = r1176296 / r1176291;
        return r1176297;
}
↓
double f(double a, double b_2, double c) {
        double r1176298 = b_2;
        double r1176299 = -3.2287897258283204e-38;
        bool r1176300 = r1176298 <= r1176299;
        double r1176301 = 0.5;
        double r1176302 = a;
        double r1176303 = c;
        double r1176304 = r1176298 / r1176303;
        double r1176305 = r1176302 / r1176304;
        double r1176306 = -2.0;
        double r1176307 = r1176298 * r1176306;
        double r1176308 = fma(r1176301, r1176305, r1176307);
        double r1176309 = r1176308 / r1176302;
        double r1176310 = 1.4954103314709843e+136;
        bool r1176311 = r1176298 <= r1176310;
        double r1176312 = r1176298 * r1176298;
        double r1176313 = r1176302 * r1176303;
        double r1176314 = r1176312 - r1176313;
        double r1176315 = sqrt(r1176314);
        double r1176316 = r1176315 + r1176298;
        double r1176317 = r1176303 / r1176316;
        double r1176318 = -r1176317;
        double r1176319 = -0.5;
        double r1176320 = r1176303 / r1176298;
        double r1176321 = r1176319 * r1176320;
        double r1176322 = r1176311 ? r1176318 : r1176321;
        double r1176323 = r1176300 ? r1176309 : r1176322;
        return r1176323;
}

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 < -3.2287897258283204e-38

    1. Initial program 28.2

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

    2. Simplified28.2

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

    3. Taylor expanded around -inf 13.9

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

    4. Simplified10.1

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

    if -3.2287897258283204e-38 < b_2 < 1.4954103314709843e+136

    1. Initial program 26.7

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

    2. Simplified26.7

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

    4. Applied div-inv26.7

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

    6. Applied flip--30.9

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

    7. Applied associate-*l/30.9

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

    8. Simplified17.6

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

    9. Taylor expanded around -inf 13.4

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

    10. Simplified13.4

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

    if 1.4954103314709843e+136 < b_2

    1. Initial program 61.3

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

    2. Simplified61.3

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

    3. Taylor expanded around inf 2.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.2287897258283204 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \mathbf{elif}\;b_2 \le 1.4954103314709843 \cdot 10^{+136}:\\ \;\;\;\;-\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))