Average Error: 1.7 → 0.4
Time: 16.6s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.071807861328125:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -0.071807861328125:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r570400 = b_2;
        double r570401 = -r570400;
        double r570402 = r570400 * r570400;
        double r570403 = a;
        double r570404 = c;
        double r570405 = r570403 * r570404;
        double r570406 = r570402 - r570405;
        double r570407 = sqrt(r570406);
        double r570408 = r570401 - r570407;
        double r570409 = r570408 / r570403;
        return r570409;
}
↓
double f(double a, double b_2, double c) {
        double r570410 = b_2;
        double r570411 = -0.071807861328125;
        bool r570412 = r570410 <= r570411;
        double r570413 = a;
        double r570414 = c;
        double r570415 = r570413 * r570414;
        double r570416 = -r570410;
        double r570417 = r570416 + r570410;
        double r570418 = r570416 + r570416;
        double r570419 = r570417 * r570418;
        double r570420 = r570415 + r570419;
        double r570421 = r570420 / r570413;
        double r570422 = r570410 * r570410;
        double r570423 = r570422 - r570415;
        double r570424 = sqrt(r570423);
        double r570425 = r570416 + r570424;
        double r570426 = r570421 / r570425;
        double r570427 = r570416 - r570424;
        double r570428 = r570427 / r570413;
        double r570429 = r570412 ? r570426 : r570428;
        return r570429;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b_2 < -0.071807861328125

    1. Initial program 3.4

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

    3. Applied p16-flip--3.0

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

    4. Applied associate-/l/3.1

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

    5. Simplified0.7

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

    7. Applied associate-/r*0.5

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

    if -0.071807861328125 < b_2

    1. Initial program 0.4

      \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.071807861328125:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))