Average Error: 32.9 → 10.5
Time: 20.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{b_2}{a}\right), -2, \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 7.055294936690956 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.1701110130378705 \cdot 10^{+68}:\\
\;\;\;\;\mathsf{fma}\left(\left(\frac{b_2}{a}\right), -2, \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\

\mathbf{elif}\;b_2 \le 7.055294936690956 \cdot 10^{-115}:\\
\;\;\;\;\frac{1}{\frac{a}{\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 r604376 = b_2;
        double r604377 = -r604376;
        double r604378 = r604376 * r604376;
        double r604379 = a;
        double r604380 = c;
        double r604381 = r604379 * r604380;
        double r604382 = r604378 - r604381;
        double r604383 = sqrt(r604382);
        double r604384 = r604377 + r604383;
        double r604385 = r604384 / r604379;
        return r604385;
}
↓
double f(double a, double b_2, double c) {
        double r604386 = b_2;
        double r604387 = -6.1701110130378705e+68;
        bool r604388 = r604386 <= r604387;
        double r604389 = a;
        double r604390 = r604386 / r604389;
        double r604391 = -2.0;
        double r604392 = 0.5;
        double r604393 = c;
        double r604394 = r604393 / r604386;
        double r604395 = r604392 * r604394;
        double r604396 = fma(r604390, r604391, r604395);
        double r604397 = 7.055294936690956e-115;
        bool r604398 = r604386 <= r604397;
        double r604399 = 1.0;
        double r604400 = r604386 * r604386;
        double r604401 = r604389 * r604393;
        double r604402 = r604400 - r604401;
        double r604403 = sqrt(r604402);
        double r604404 = r604403 - r604386;
        double r604405 = r604389 / r604404;
        double r604406 = r604399 / r604405;
        double r604407 = -0.5;
        double r604408 = r604407 * r604394;
        double r604409 = r604398 ? r604406 : r604408;
        double r604410 = r604388 ? r604396 : r604409;
        return r604410;
}

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 < -6.1701110130378705e+68

    1. Initial program 38.0

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

    2. Simplified38.0

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

    3. Taylor expanded around -inf 4.7

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

    4. Simplified4.7

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

    if -6.1701110130378705e+68 < b_2 < 7.055294936690956e-115

    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

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

    if 7.055294936690956e-115 < b_2

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around 0 50.8

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

    4. Simplified50.8

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

    5. Taylor expanded around inf 11.5

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{b_2}{a}\right), -2, \left(\frac{1}{2} \cdot \frac{c}{b_2}\right)\right)\\ \mathbf{elif}\;b_2 \le 7.055294936690956 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +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))