Average Error: 28.6 → 16.6
Time: 18.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 77.85350311081474217189679620787501335144:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 77.85350311081474217189679620787501335144:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2365318 = b;
        double r2365319 = -r2365318;
        double r2365320 = r2365318 * r2365318;
        double r2365321 = 4.0;
        double r2365322 = a;
        double r2365323 = r2365321 * r2365322;
        double r2365324 = c;
        double r2365325 = r2365323 * r2365324;
        double r2365326 = r2365320 - r2365325;
        double r2365327 = sqrt(r2365326);
        double r2365328 = r2365319 + r2365327;
        double r2365329 = 2.0;
        double r2365330 = r2365329 * r2365322;
        double r2365331 = r2365328 / r2365330;
        return r2365331;
}
↓
double f(double a, double b, double c) {
        double r2365332 = b;
        double r2365333 = 77.85350311081474;
        bool r2365334 = r2365332 <= r2365333;
        double r2365335 = r2365332 * r2365332;
        double r2365336 = a;
        double r2365337 = 4.0;
        double r2365338 = r2365336 * r2365337;
        double r2365339 = c;
        double r2365340 = r2365338 * r2365339;
        double r2365341 = r2365335 - r2365340;
        double r2365342 = sqrt(r2365341);
        double r2365343 = r2365341 * r2365342;
        double r2365344 = r2365335 * r2365332;
        double r2365345 = r2365343 - r2365344;
        double r2365346 = r2365332 * r2365342;
        double r2365347 = r2365346 + r2365335;
        double r2365348 = r2365341 + r2365347;
        double r2365349 = r2365345 / r2365348;
        double r2365350 = r2365349 / r2365336;
        double r2365351 = 2.0;
        double r2365352 = r2365350 / r2365351;
        double r2365353 = -2.0;
        double r2365354 = r2365339 / r2365332;
        double r2365355 = r2365353 * r2365354;
        double r2365356 = r2365355 / r2365351;
        double r2365357 = r2365334 ? r2365352 : r2365356;
        return r2365357;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 77.85350311081474

    1. Initial program 14.9

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

    2. Simplified14.9

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

    4. Applied flip3--15.0

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

    5. Simplified14.3

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

    6. Simplified14.3

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

    if 77.85350311081474 < b

    1. Initial program 34.5

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

    2. Simplified34.5

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

    3. Taylor expanded around inf 17.6

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 77.85350311081474217189679620787501335144:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(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) (* (* 4.0 a) c)))) (* 2.0 a)))