Average Error: 43.7 → 11.2
Time: 18.8s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.002027355084314286:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 0.002027355084314286:\\
\;\;\;\;\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(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b, 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 r983056 = b;
        double r983057 = -r983056;
        double r983058 = r983056 * r983056;
        double r983059 = 4.0;
        double r983060 = a;
        double r983061 = r983059 * r983060;
        double r983062 = c;
        double r983063 = r983061 * r983062;
        double r983064 = r983058 - r983063;
        double r983065 = sqrt(r983064);
        double r983066 = r983057 + r983065;
        double r983067 = 2.0;
        double r983068 = r983067 * r983060;
        double r983069 = r983066 / r983068;
        return r983069;
}
↓
double f(double a, double b, double c) {
        double r983070 = b;
        double r983071 = 0.002027355084314286;
        bool r983072 = r983070 <= r983071;
        double r983073 = a;
        double r983074 = c;
        double r983075 = r983073 * r983074;
        double r983076 = -4.0;
        double r983077 = r983070 * r983070;
        double r983078 = fma(r983075, r983076, r983077);
        double r983079 = sqrt(r983078);
        double r983080 = r983079 * r983078;
        double r983081 = r983077 * r983070;
        double r983082 = r983080 - r983081;
        double r983083 = r983077 + r983078;
        double r983084 = fma(r983079, r983070, r983083);
        double r983085 = r983082 / r983084;
        double r983086 = r983085 / r983073;
        double r983087 = 2.0;
        double r983088 = r983086 / r983087;
        double r983089 = -2.0;
        double r983090 = r983074 / r983070;
        double r983091 = r983089 * r983090;
        double r983092 = r983091 / r983087;
        double r983093 = r983072 ? r983088 : r983092;
        return r983093;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.002027355084314286

    1. Initial program 20.3

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

    2. Simplified20.2

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

    4. Applied flip3--20.4

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

    5. Simplified19.8

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

    6. Simplified19.8

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

    if 0.002027355084314286 < b

    1. Initial program 46.1

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

    2. Simplified46.1

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

    3. Taylor expanded around inf 10.3

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.002027355084314286:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))