Average Error: 28.5 → 16.2
Time: 15.0s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 2495.5039318207096:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r659515 = b;
        double r659516 = -r659515;
        double r659517 = r659515 * r659515;
        double r659518 = 4.0;
        double r659519 = a;
        double r659520 = r659518 * r659519;
        double r659521 = c;
        double r659522 = r659520 * r659521;
        double r659523 = r659517 - r659522;
        double r659524 = sqrt(r659523);
        double r659525 = r659516 + r659524;
        double r659526 = 2.0;
        double r659527 = r659526 * r659519;
        double r659528 = r659525 / r659527;
        return r659528;
}
↓
double f(double a, double b, double c) {
        double r659529 = b;
        double r659530 = 2495.5039318207096;
        bool r659531 = r659529 <= r659530;
        double r659532 = r659529 * r659529;
        double r659533 = a;
        double r659534 = c;
        double r659535 = r659533 * r659534;
        double r659536 = 4.0;
        double r659537 = r659535 * r659536;
        double r659538 = r659532 - r659537;
        double r659539 = sqrt(r659538);
        double r659540 = r659538 * r659539;
        double r659541 = r659532 * r659529;
        double r659542 = r659540 - r659541;
        double r659543 = r659529 + r659539;
        double r659544 = r659529 * r659543;
        double r659545 = r659538 + r659544;
        double r659546 = r659542 / r659545;
        double r659547 = r659546 / r659533;
        double r659548 = 2.0;
        double r659549 = r659547 / r659548;
        double r659550 = -2.0;
        double r659551 = r659534 / r659529;
        double r659552 = r659550 * r659551;
        double r659553 = r659552 / r659548;
        double r659554 = r659531 ? r659549 : r659553;
        return r659554;
}

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 < 2495.5039318207096

    1. Initial program 17.8

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

    2. Simplified17.8

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

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

    6. Simplified17.2

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

    if 2495.5039318207096 < b

    1. Initial program 37.3

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

    2. Simplified37.3

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

    3. Taylor expanded around inf 15.5

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))