Average Error: 43.6 → 11.3
Time: 14.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0015483000136869431:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 0.0015483000136869431:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, 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 r513318 = b;
        double r513319 = -r513318;
        double r513320 = r513318 * r513318;
        double r513321 = 4.0;
        double r513322 = a;
        double r513323 = r513321 * r513322;
        double r513324 = c;
        double r513325 = r513323 * r513324;
        double r513326 = r513320 - r513325;
        double r513327 = sqrt(r513326);
        double r513328 = r513319 + r513327;
        double r513329 = 2.0;
        double r513330 = r513329 * r513322;
        double r513331 = r513328 / r513330;
        return r513331;
}
↓
double f(double a, double b, double c) {
        double r513332 = b;
        double r513333 = 0.0015483000136869431;
        bool r513334 = r513332 <= r513333;
        double r513335 = c;
        double r513336 = -4.0;
        double r513337 = r513335 * r513336;
        double r513338 = a;
        double r513339 = r513332 * r513332;
        double r513340 = fma(r513337, r513338, r513339);
        double r513341 = sqrt(r513340);
        double r513342 = r513341 * r513340;
        double r513343 = r513339 * r513332;
        double r513344 = r513342 - r513343;
        double r513345 = r513339 + r513340;
        double r513346 = fma(r513332, r513341, r513345);
        double r513347 = r513344 / r513346;
        double r513348 = r513347 / r513338;
        double r513349 = 2.0;
        double r513350 = r513348 / r513349;
        double r513351 = -2.0;
        double r513352 = r513335 / r513332;
        double r513353 = r513351 * r513352;
        double r513354 = r513353 / r513349;
        double r513355 = r513334 ? r513350 : r513354;
        return r513355;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.0015483000136869431

    1. Initial program 19.9

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

    2. Simplified19.9

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

    4. Applied flip3--19.9

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

    5. Simplified19.2

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

    6. Simplified19.2

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

    if 0.0015483000136869431 < b

    1. Initial program 45.8

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0015483000136869431:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))