Average Error: 28.1 → 16.3
Time: 36.6s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1315.7127116390275:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 1315.7127116390275:\\
\;\;\;\;\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 r1066025 = b;
        double r1066026 = -r1066025;
        double r1066027 = r1066025 * r1066025;
        double r1066028 = 4.0;
        double r1066029 = a;
        double r1066030 = r1066028 * r1066029;
        double r1066031 = c;
        double r1066032 = r1066030 * r1066031;
        double r1066033 = r1066027 - r1066032;
        double r1066034 = sqrt(r1066033);
        double r1066035 = r1066026 + r1066034;
        double r1066036 = 2.0;
        double r1066037 = r1066036 * r1066029;
        double r1066038 = r1066035 / r1066037;
        return r1066038;
}
↓
double f(double a, double b, double c) {
        double r1066039 = b;
        double r1066040 = 1315.7127116390275;
        bool r1066041 = r1066039 <= r1066040;
        double r1066042 = a;
        double r1066043 = c;
        double r1066044 = r1066042 * r1066043;
        double r1066045 = -4.0;
        double r1066046 = r1066039 * r1066039;
        double r1066047 = fma(r1066044, r1066045, r1066046);
        double r1066048 = sqrt(r1066047);
        double r1066049 = r1066048 * r1066047;
        double r1066050 = r1066046 * r1066039;
        double r1066051 = r1066049 - r1066050;
        double r1066052 = r1066046 + r1066047;
        double r1066053 = fma(r1066039, r1066048, r1066052);
        double r1066054 = r1066051 / r1066053;
        double r1066055 = r1066054 / r1066042;
        double r1066056 = 2.0;
        double r1066057 = r1066055 / r1066056;
        double r1066058 = -2.0;
        double r1066059 = r1066043 / r1066039;
        double r1066060 = r1066058 * r1066059;
        double r1066061 = r1066060 / r1066056;
        double r1066062 = r1066041 ? r1066057 : r1066061;
        return r1066062;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1315.7127116390275

    1. Initial program 16.9

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

    2. Simplified16.8

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

    4. Applied flip3--16.8

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

    5. Simplified16.2

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

    6. Simplified16.2

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

    if 1315.7127116390275 < b

    1. Initial program 36.2

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 16.4

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1315.7127116390275:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +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)))