Average Error: 28.7 → 16.1
Time: 17.7s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 315.4848387613182580935244914144277572632:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(b \cdot b + c \cdot \left(4 \cdot a\right)\right)}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]
\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 315.4848387613182580935244914144277572632:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \left(b \cdot b + c \cdot \left(4 \cdot a\right)\right)}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + b}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c \cdot -1}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r39486 = b;
        double r39487 = -r39486;
        double r39488 = r39486 * r39486;
        double r39489 = 4.0;
        double r39490 = a;
        double r39491 = r39489 * r39490;
        double r39492 = c;
        double r39493 = r39491 * r39492;
        double r39494 = r39488 - r39493;
        double r39495 = sqrt(r39494);
        double r39496 = r39487 + r39495;
        double r39497 = 2.0;
        double r39498 = r39497 * r39490;
        double r39499 = r39496 / r39498;
        return r39499;
}
↓
double f(double a, double b, double c) {
        double r39500 = b;
        double r39501 = 315.48483876131826;
        bool r39502 = r39500 <= r39501;
        double r39503 = r39500 * r39500;
        double r39504 = c;
        double r39505 = 4.0;
        double r39506 = a;
        double r39507 = r39505 * r39506;
        double r39508 = r39504 * r39507;
        double r39509 = r39503 + r39508;
        double r39510 = r39503 - r39509;
        double r39511 = r39503 - r39508;
        double r39512 = sqrt(r39511);
        double r39513 = r39512 + r39500;
        double r39514 = r39510 / r39513;
        double r39515 = 2.0;
        double r39516 = r39514 / r39515;
        double r39517 = r39516 / r39506;
        double r39518 = -1.0;
        double r39519 = r39504 * r39518;
        double r39520 = r39519 / r39500;
        double r39521 = r39502 ? r39517 : r39520;
        return r39521;
}

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 < 315.48483876131826

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied flip--16.3

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

    5. Simplified15.3

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

    6. Simplified15.3

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

    if 315.48483876131826 < b

    1. Initial program 35.8

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 16.7

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

    4. Simplified16.7

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

    5. Taylor expanded around 0 16.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    6. Simplified16.6

      \[\leadsto \color{blue}{\frac{c \cdot -1}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 315.4848387613182580935244914144277572632:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(b \cdot b + c \cdot \left(4 \cdot a\right)\right)}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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)))