Average Error: 28.8 → 16.1
Time: 20.4s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 627.9389082699839:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 627.9389082699839:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1169469 = b;
        double r1169470 = -r1169469;
        double r1169471 = r1169469 * r1169469;
        double r1169472 = 4.0;
        double r1169473 = a;
        double r1169474 = r1169472 * r1169473;
        double r1169475 = c;
        double r1169476 = r1169474 * r1169475;
        double r1169477 = r1169471 - r1169476;
        double r1169478 = sqrt(r1169477);
        double r1169479 = r1169470 + r1169478;
        double r1169480 = 2.0;
        double r1169481 = r1169480 * r1169473;
        double r1169482 = r1169479 / r1169481;
        return r1169482;
}
↓
double f(double a, double b, double c) {
        double r1169483 = b;
        double r1169484 = 627.9389082699839;
        bool r1169485 = r1169483 <= r1169484;
        double r1169486 = r1169483 * r1169483;
        double r1169487 = c;
        double r1169488 = a;
        double r1169489 = r1169487 * r1169488;
        double r1169490 = -4.0;
        double r1169491 = r1169489 * r1169490;
        double r1169492 = r1169486 + r1169491;
        double r1169493 = sqrt(r1169492);
        double r1169494 = r1169492 * r1169493;
        double r1169495 = r1169486 * r1169483;
        double r1169496 = r1169494 - r1169495;
        double r1169497 = r1169483 + r1169493;
        double r1169498 = r1169497 * r1169493;
        double r1169499 = r1169498 + r1169486;
        double r1169500 = r1169496 / r1169499;
        double r1169501 = r1169500 / r1169488;
        double r1169502 = 2.0;
        double r1169503 = r1169501 / r1169502;
        double r1169504 = -2.0;
        double r1169505 = r1169487 / r1169483;
        double r1169506 = r1169504 * r1169505;
        double r1169507 = r1169506 / r1169502;
        double r1169508 = r1169485 ? r1169503 : r1169507;
        return r1169508;
}

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 < 627.9389082699839

    1. Initial program 16.6

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

    2. Simplified16.6

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

    4. Applied flip3--16.6

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

    5. Simplified15.9

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

    6. Simplified15.9

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

    if 627.9389082699839 < b

    1. Initial program 36.4

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

    2. Simplified36.4

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

    3. Taylor expanded around inf 16.2

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 627.9389082699839:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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)))