Average Error: 29.0 → 16.2
Time: 14.2s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 759.6594316796017:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 759.6594316796017:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r736484 = b;
        double r736485 = -r736484;
        double r736486 = r736484 * r736484;
        double r736487 = 4.0;
        double r736488 = a;
        double r736489 = r736487 * r736488;
        double r736490 = c;
        double r736491 = r736489 * r736490;
        double r736492 = r736486 - r736491;
        double r736493 = sqrt(r736492);
        double r736494 = r736485 + r736493;
        double r736495 = 2.0;
        double r736496 = r736495 * r736488;
        double r736497 = r736494 / r736496;
        return r736497;
}
↓
double f(double a, double b, double c) {
        double r736498 = b;
        double r736499 = 759.6594316796017;
        bool r736500 = r736498 <= r736499;
        double r736501 = r736498 * r736498;
        double r736502 = 4.0;
        double r736503 = c;
        double r736504 = a;
        double r736505 = r736503 * r736504;
        double r736506 = r736502 * r736505;
        double r736507 = r736501 - r736506;
        double r736508 = r736507 - r736501;
        double r736509 = sqrt(r736507);
        double r736510 = r736498 + r736509;
        double r736511 = r736508 / r736510;
        double r736512 = r736511 / r736504;
        double r736513 = 2.0;
        double r736514 = r736512 / r736513;
        double r736515 = r736503 / r736498;
        double r736516 = -2.0;
        double r736517 = r736515 * r736516;
        double r736518 = r736517 / r736513;
        double r736519 = r736500 ? r736514 : r736518;
        return r736519;
}

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 < 759.6594316796017

    1. Initial program 17.3

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

    2. Simplified17.3

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

    4. Applied flip--17.3

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

    5. Simplified16.2

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

    if 759.6594316796017 < b

    1. Initial program 36.5

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

    2. Simplified36.5

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

    3. Taylor expanded around inf 16.1

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 759.6594316796017:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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)))