Average Error: 33.7 → 10.3
Time: 26.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r941223 = b_2;
        double r941224 = -r941223;
        double r941225 = r941223 * r941223;
        double r941226 = a;
        double r941227 = c;
        double r941228 = r941226 * r941227;
        double r941229 = r941225 - r941228;
        double r941230 = sqrt(r941229);
        double r941231 = r941224 + r941230;
        double r941232 = r941231 / r941226;
        return r941232;
}
↓
double f(double a, double b_2, double c) {
        double r941233 = b_2;
        double r941234 = -2.0256498248166784e+153;
        bool r941235 = r941233 <= r941234;
        double r941236 = a;
        double r941237 = r941233 / r941236;
        double r941238 = -2.0;
        double r941239 = 0.5;
        double r941240 = c;
        double r941241 = r941240 / r941233;
        double r941242 = r941239 * r941241;
        double r941243 = fma(r941237, r941238, r941242);
        double r941244 = 3.0476772566360775e-81;
        bool r941245 = r941233 <= r941244;
        double r941246 = r941233 * r941233;
        double r941247 = r941236 * r941240;
        double r941248 = r941246 - r941247;
        double r941249 = sqrt(r941248);
        double r941250 = r941249 / r941236;
        double r941251 = r941250 - r941237;
        double r941252 = -0.5;
        double r941253 = r941241 * r941252;
        double r941254 = r941245 ? r941251 : r941253;
        double r941255 = r941235 ? r941243 : r941254;
        return r941255;
}

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 < -2.0256498248166784e+153

    1. Initial program 63.6

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

    2. Simplified63.6

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

    4. Applied div-sub63.6

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

    5. Taylor expanded around -inf 2.0

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

    6. Simplified2.0

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

    if -2.0256498248166784e+153 < b_2 < 3.0476772566360775e-81

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied div-sub11.9

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

    if 3.0476772566360775e-81 < b_2

    1. Initial program 52.1

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

    2. Simplified52.1

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

    3. Taylor expanded around inf 10.5

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))