Average Error: 28.8 → 16.2
Time: 18.5s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2650.8680979433148:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 2650.8680979433148:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1114439 = b;
        double r1114440 = -r1114439;
        double r1114441 = r1114439 * r1114439;
        double r1114442 = 4.0;
        double r1114443 = a;
        double r1114444 = r1114442 * r1114443;
        double r1114445 = c;
        double r1114446 = r1114444 * r1114445;
        double r1114447 = r1114441 - r1114446;
        double r1114448 = sqrt(r1114447);
        double r1114449 = r1114440 + r1114448;
        double r1114450 = 2.0;
        double r1114451 = r1114450 * r1114443;
        double r1114452 = r1114449 / r1114451;
        return r1114452;
}
↓
double f(double a, double b, double c) {
        double r1114453 = b;
        double r1114454 = 2650.8680979433148;
        bool r1114455 = r1114453 <= r1114454;
        double r1114456 = a;
        double r1114457 = c;
        double r1114458 = r1114456 * r1114457;
        double r1114459 = -4.0;
        double r1114460 = r1114453 * r1114453;
        double r1114461 = fma(r1114458, r1114459, r1114460);
        double r1114462 = sqrt(r1114461);
        double r1114463 = r1114462 * r1114461;
        double r1114464 = r1114460 * r1114453;
        double r1114465 = r1114463 - r1114464;
        double r1114466 = r1114460 + r1114461;
        double r1114467 = fma(r1114453, r1114462, r1114466);
        double r1114468 = r1114465 / r1114467;
        double r1114469 = r1114468 / r1114456;
        double r1114470 = 2.0;
        double r1114471 = r1114469 / r1114470;
        double r1114472 = -2.0;
        double r1114473 = r1114457 / r1114453;
        double r1114474 = r1114472 * r1114473;
        double r1114475 = r1114474 / r1114470;
        double r1114476 = r1114455 ? r1114471 : r1114475;
        return r1114476;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2650.8680979433148

    1. Initial program 18.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified18.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied flip3--18.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}\right)}^{3} - {b}^{3}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + \left(b \cdot b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot b\right)}}}{a}}{2}\]

    5. Simplified17.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - b \cdot \left(b \cdot b\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} + \left(b \cdot b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} \cdot b\right)}}{a}}{2}\]

    6. Simplified17.4

      \[\leadsto \frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - b \cdot \left(b \cdot b\right)}{\color{blue}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) + b \cdot b\right)}}}{a}}{2}\]

    if 2650.8680979433148 < b

    1. Initial program 37.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified37.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b}{a}}{2}}\]

    3. Taylor expanded around inf 15.3

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2650.8680979433148:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))