Average Error: 28.4 → 16.4
Time: 18.2s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2524.176906759875:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\sqrt{a}} \cdot \frac{\frac{a}{\frac{b}{c}}}{\sqrt{a}}}{2}\\ \end{array}\]
\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 2524.176906759875:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1365326 = b;
        double r1365327 = -r1365326;
        double r1365328 = r1365326 * r1365326;
        double r1365329 = 4.0;
        double r1365330 = a;
        double r1365331 = r1365329 * r1365330;
        double r1365332 = c;
        double r1365333 = r1365331 * r1365332;
        double r1365334 = r1365328 - r1365333;
        double r1365335 = sqrt(r1365334);
        double r1365336 = r1365327 + r1365335;
        double r1365337 = 2.0;
        double r1365338 = r1365337 * r1365330;
        double r1365339 = r1365336 / r1365338;
        return r1365339;
}
↓
double f(double a, double b, double c) {
        double r1365340 = b;
        double r1365341 = 2524.176906759875;
        bool r1365342 = r1365340 <= r1365341;
        double r1365343 = r1365340 * r1365340;
        double r1365344 = -4.0;
        double r1365345 = a;
        double r1365346 = r1365344 * r1365345;
        double r1365347 = c;
        double r1365348 = r1365346 * r1365347;
        double r1365349 = r1365343 + r1365348;
        double r1365350 = sqrt(r1365349);
        double r1365351 = r1365349 * r1365350;
        double r1365352 = r1365343 * r1365340;
        double r1365353 = r1365351 - r1365352;
        double r1365354 = r1365340 * r1365350;
        double r1365355 = r1365354 + r1365343;
        double r1365356 = r1365349 + r1365355;
        double r1365357 = r1365353 / r1365356;
        double r1365358 = r1365357 / r1365345;
        double r1365359 = 2.0;
        double r1365360 = r1365358 / r1365359;
        double r1365361 = -2.0;
        double r1365362 = sqrt(r1365345);
        double r1365363 = r1365361 / r1365362;
        double r1365364 = r1365340 / r1365347;
        double r1365365 = r1365345 / r1365364;
        double r1365366 = r1365365 / r1365362;
        double r1365367 = r1365363 * r1365366;
        double r1365368 = r1365367 / r1365359;
        double r1365369 = r1365342 ? r1365360 : r1365368;
        return r1365369;
}

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 < 2524.176906759875

    1. Initial program 17.7

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

    2. Simplified17.7

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

    4. Applied flip3--17.9

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

    5. Simplified17.2

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

    6. Simplified17.2

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

    if 2524.176906759875 < b

    1. Initial program 37.2

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

    2. Simplified37.2

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

    3. Taylor expanded around inf 15.7

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a}}{2}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt15.8

      \[\leadsto \frac{\frac{-2 \cdot \frac{a \cdot c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}}{a}}{2}\]

    6. Applied associate-/r*15.8

      \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\frac{\frac{a \cdot c}{\sqrt{b}}}{\sqrt{b}}}}{a}}{2}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt15.8

      \[\leadsto \frac{\frac{-2 \cdot \frac{\frac{a \cdot c}{\sqrt{b}}}{\sqrt{b}}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}}{2}\]

    9. Applied times-frac15.8

      \[\leadsto \frac{\color{blue}{\frac{-2}{\sqrt{a}} \cdot \frac{\frac{\frac{a \cdot c}{\sqrt{b}}}{\sqrt{b}}}{\sqrt{a}}}}{2}\]

    10. Simplified15.8

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2524.176906759875:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(-4 \cdot a\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\sqrt{a}} \cdot \frac{\frac{a}{\frac{b}{c}}}{\sqrt{a}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))