Average Error: 28.4 → 16.9
Time: 15.6s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 6094.521206369936:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 6094.521206369936:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b \cdot b\right)}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1011367 = b;
        double r1011368 = -r1011367;
        double r1011369 = r1011367 * r1011367;
        double r1011370 = 4.0;
        double r1011371 = a;
        double r1011372 = r1011370 * r1011371;
        double r1011373 = c;
        double r1011374 = r1011372 * r1011373;
        double r1011375 = r1011369 - r1011374;
        double r1011376 = sqrt(r1011375);
        double r1011377 = r1011368 + r1011376;
        double r1011378 = 2.0;
        double r1011379 = r1011378 * r1011371;
        double r1011380 = r1011377 / r1011379;
        return r1011380;
}
↓
double f(double a, double b, double c) {
        double r1011381 = b;
        double r1011382 = 6094.521206369936;
        bool r1011383 = r1011381 <= r1011382;
        double r1011384 = r1011381 * r1011381;
        double r1011385 = 4.0;
        double r1011386 = c;
        double r1011387 = a;
        double r1011388 = r1011386 * r1011387;
        double r1011389 = r1011385 * r1011388;
        double r1011390 = r1011384 - r1011389;
        double r1011391 = sqrt(r1011390);
        double r1011392 = r1011390 * r1011391;
        double r1011393 = r1011384 * r1011381;
        double r1011394 = r1011392 - r1011393;
        double r1011395 = r1011381 * r1011391;
        double r1011396 = r1011395 + r1011384;
        double r1011397 = r1011390 + r1011396;
        double r1011398 = r1011394 / r1011397;
        double r1011399 = 2.0;
        double r1011400 = r1011387 * r1011399;
        double r1011401 = r1011398 / r1011400;
        double r1011402 = -r1011386;
        double r1011403 = r1011402 / r1011381;
        double r1011404 = r1011383 ? r1011401 : r1011403;
        return r1011404;
}

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 < 6094.521206369936

    1. Initial program 19.2

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

    2. Simplified19.2

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

    4. Applied flip3--19.2

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

    5. Simplified18.5

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

    6. Simplified18.5

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

    if 6094.521206369936 < 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

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

    3. Taylor expanded around inf 15.4

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

    4. Simplified15.4

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

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6094.521206369936:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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)))