Average Error: 28.5 → 16.7
Time: 14.7s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 2892.1913455639924:\\
\;\;\;\;\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 r656212 = b;
        double r656213 = -r656212;
        double r656214 = r656212 * r656212;
        double r656215 = 4.0;
        double r656216 = a;
        double r656217 = r656215 * r656216;
        double r656218 = c;
        double r656219 = r656217 * r656218;
        double r656220 = r656214 - r656219;
        double r656221 = sqrt(r656220);
        double r656222 = r656213 + r656221;
        double r656223 = 2.0;
        double r656224 = r656223 * r656216;
        double r656225 = r656222 / r656224;
        return r656225;
}
↓
double f(double a, double b, double c) {
        double r656226 = b;
        double r656227 = 2892.1913455639924;
        bool r656228 = r656226 <= r656227;
        double r656229 = c;
        double r656230 = -4.0;
        double r656231 = r656229 * r656230;
        double r656232 = a;
        double r656233 = r656226 * r656226;
        double r656234 = fma(r656231, r656232, r656233);
        double r656235 = sqrt(r656234);
        double r656236 = r656235 * r656234;
        double r656237 = r656233 * r656226;
        double r656238 = r656236 - r656237;
        double r656239 = r656233 + r656234;
        double r656240 = fma(r656226, r656235, r656239);
        double r656241 = r656238 / r656240;
        double r656242 = r656241 / r656232;
        double r656243 = 2.0;
        double r656244 = r656242 / r656243;
        double r656245 = -2.0;
        double r656246 = r656229 / r656226;
        double r656247 = r656245 * r656246;
        double r656248 = r656247 / r656243;
        double r656249 = r656228 ? r656244 : r656248;
        return r656249;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2892.1913455639924

    1. Initial program 18.5

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

    2. Simplified18.4

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

    4. Applied flip3--18.5

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

    5. Simplified17.8

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

    6. Simplified17.8

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

    if 2892.1913455639924 < b

    1. Initial program 36.9

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

    2. Simplified36.9

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

    3. Taylor expanded around inf 15.7

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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)))