Average Error: 44.0 → 11.2
Time: 18.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.12005985553413497:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 0.12005985553413497:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(-4, a \cdot c, 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 r1192757 = b;
        double r1192758 = -r1192757;
        double r1192759 = r1192757 * r1192757;
        double r1192760 = 4.0;
        double r1192761 = a;
        double r1192762 = r1192760 * r1192761;
        double r1192763 = c;
        double r1192764 = r1192762 * r1192763;
        double r1192765 = r1192759 - r1192764;
        double r1192766 = sqrt(r1192765);
        double r1192767 = r1192758 + r1192766;
        double r1192768 = 2.0;
        double r1192769 = r1192768 * r1192761;
        double r1192770 = r1192767 / r1192769;
        return r1192770;
}
↓
double f(double a, double b, double c) {
        double r1192771 = b;
        double r1192772 = 0.12005985553413497;
        bool r1192773 = r1192771 <= r1192772;
        double r1192774 = -4.0;
        double r1192775 = a;
        double r1192776 = c;
        double r1192777 = r1192775 * r1192776;
        double r1192778 = r1192771 * r1192771;
        double r1192779 = fma(r1192774, r1192777, r1192778);
        double r1192780 = sqrt(r1192779);
        double r1192781 = r1192780 * r1192779;
        double r1192782 = r1192778 * r1192771;
        double r1192783 = r1192781 - r1192782;
        double r1192784 = r1192778 + r1192779;
        double r1192785 = fma(r1192780, r1192771, r1192784);
        double r1192786 = r1192783 / r1192785;
        double r1192787 = r1192786 / r1192775;
        double r1192788 = 2.0;
        double r1192789 = r1192787 / r1192788;
        double r1192790 = -2.0;
        double r1192791 = r1192776 / r1192771;
        double r1192792 = r1192790 * r1192791;
        double r1192793 = r1192792 / r1192788;
        double r1192794 = r1192773 ? r1192789 : r1192793;
        return r1192794;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.12005985553413497

    1. Initial program 23.2

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

    2. Simplified23.2

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

    4. Applied flip3--23.3

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

    5. Simplified22.7

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

    6. Simplified22.7

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

    if 0.12005985553413497 < b

    1. Initial program 47.2

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

    2. Simplified47.2

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.12005985553413497:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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)))