Average Error: 33.8 → 10.3
Time: 17.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\ \mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]
\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 -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\
\;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\

\mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r41366 = b;
        double r41367 = -r41366;
        double r41368 = r41366 * r41366;
        double r41369 = 4.0;
        double r41370 = a;
        double r41371 = r41369 * r41370;
        double r41372 = c;
        double r41373 = r41371 * r41372;
        double r41374 = r41368 - r41373;
        double r41375 = sqrt(r41374);
        double r41376 = r41367 + r41375;
        double r41377 = 2.0;
        double r41378 = r41377 * r41370;
        double r41379 = r41376 / r41378;
        return r41379;
}
↓
double f(double a, double b, double c) {
        double r41380 = b;
        double r41381 = -2.0256498248166784e+153;
        bool r41382 = r41380 <= r41381;
        double r41383 = c;
        double r41384 = 1.0;
        double r41385 = r41383 * r41384;
        double r41386 = r41385 / r41380;
        double r41387 = 0.5;
        double r41388 = a;
        double r41389 = r41380 / r41388;
        double r41390 = r41387 * r41389;
        double r41391 = r41386 - r41390;
        double r41392 = 2.0;
        double r41393 = r41380 / r41392;
        double r41394 = r41393 / r41388;
        double r41395 = r41391 - r41394;
        double r41396 = 3.0476772566360775e-81;
        bool r41397 = r41380 <= r41396;
        double r41398 = r41380 * r41380;
        double r41399 = 4.0;
        double r41400 = r41383 * r41388;
        double r41401 = r41399 * r41400;
        double r41402 = r41398 - r41401;
        double r41403 = sqrt(r41402);
        double r41404 = r41392 * r41388;
        double r41405 = r41403 / r41404;
        double r41406 = r41405 - r41394;
        double r41407 = -1.0;
        double r41408 = r41383 * r41407;
        double r41409 = r41408 / r41380;
        double r41410 = r41397 ? r41406 : r41409;
        double r41411 = r41382 ? r41395 : r41410;
        return r41411;
}

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 3 regimes

  2. if b < -2.0256498248166784e+153

    1. Initial program 63.6

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

    2. Simplified63.6

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

    4. Applied div-sub63.6

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

    5. Applied div-sub63.6

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

    6. Simplified63.6

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

    7. Taylor expanded around -inf 2.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{c}{b} - 0.5 \cdot \frac{b}{a}\right)} - \frac{\frac{b}{2}}{a}\]

    8. Simplified2.0

      \[\leadsto \color{blue}{\left(\frac{c \cdot 1}{b} - \frac{b}{a} \cdot 0.5\right)} - \frac{\frac{b}{2}}{a}\]

    if -2.0256498248166784e+153 < b < 3.0476772566360775e-81

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied div-sub11.9

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

    5. Applied div-sub11.9

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

    6. Simplified11.9

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

    if 3.0476772566360775e-81 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.5

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

    4. Simplified10.5

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.025649824816678368861606895534923213042 \cdot 10^{153}:\\ \;\;\;\;\left(\frac{c \cdot 1}{b} - 0.5 \cdot \frac{b}{a}\right) - \frac{\frac{b}{2}}{a}\\ \mathbf{elif}\;b \le 3.047677256636077515553757160900796353717 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))