Average Error: 43.8 → 11.2
Time: 18.7s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.1979590906586723:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 0.1979590906586723:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1170838 = b;
        double r1170839 = -r1170838;
        double r1170840 = r1170838 * r1170838;
        double r1170841 = 4.0;
        double r1170842 = a;
        double r1170843 = r1170841 * r1170842;
        double r1170844 = c;
        double r1170845 = r1170843 * r1170844;
        double r1170846 = r1170840 - r1170845;
        double r1170847 = sqrt(r1170846);
        double r1170848 = r1170839 + r1170847;
        double r1170849 = 2.0;
        double r1170850 = r1170849 * r1170842;
        double r1170851 = r1170848 / r1170850;
        return r1170851;
}
↓
double f(double a, double b, double c) {
        double r1170852 = b;
        double r1170853 = 0.1979590906586723;
        bool r1170854 = r1170852 <= r1170853;
        double r1170855 = c;
        double r1170856 = -4.0;
        double r1170857 = a;
        double r1170858 = r1170856 * r1170857;
        double r1170859 = r1170855 * r1170858;
        double r1170860 = fma(r1170852, r1170852, r1170859);
        double r1170861 = r1170852 * r1170852;
        double r1170862 = r1170860 - r1170861;
        double r1170863 = sqrt(r1170860);
        double r1170864 = r1170852 + r1170863;
        double r1170865 = r1170862 / r1170864;
        double r1170866 = 2.0;
        double r1170867 = r1170865 / r1170866;
        double r1170868 = r1170867 / r1170857;
        double r1170869 = -r1170855;
        double r1170870 = r1170869 / r1170852;
        double r1170871 = r1170854 ? r1170868 : r1170870;
        return r1170871;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.1979590906586723

    1. Initial program 22.9

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

    2. Simplified22.8

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

    4. Applied flip--22.9

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

    5. Simplified22.1

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

    if 0.1979590906586723 < b

    1. Initial program 47.4

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

    2. Simplified47.3

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

    3. Taylor expanded around inf 9.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    4. Simplified9.4

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.1979590906586723:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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)))