Average Error: 33.8 → 6.7
Time: 15.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -7.89283599384243582562951448457087501988 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -7.89283599384243582562951448457087501988 \cdot 10^{-230}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\
\;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r50288 = b;
        double r50289 = -r50288;
        double r50290 = r50288 * r50288;
        double r50291 = 4.0;
        double r50292 = a;
        double r50293 = r50291 * r50292;
        double r50294 = c;
        double r50295 = r50293 * r50294;
        double r50296 = r50290 - r50295;
        double r50297 = sqrt(r50296);
        double r50298 = r50289 + r50297;
        double r50299 = 2.0;
        double r50300 = r50299 * r50292;
        double r50301 = r50298 / r50300;
        return r50301;
}
↓
double f(double a, double b, double c) {
        double r50302 = b;
        double r50303 = -8.301687926884189e+98;
        bool r50304 = r50302 <= r50303;
        double r50305 = 1.0;
        double r50306 = c;
        double r50307 = r50306 / r50302;
        double r50308 = a;
        double r50309 = r50302 / r50308;
        double r50310 = r50307 - r50309;
        double r50311 = r50305 * r50310;
        double r50312 = -7.892835993842436e-230;
        bool r50313 = r50302 <= r50312;
        double r50314 = -r50302;
        double r50315 = r50302 * r50302;
        double r50316 = 4.0;
        double r50317 = r50316 * r50308;
        double r50318 = r50317 * r50306;
        double r50319 = r50315 - r50318;
        double r50320 = sqrt(r50319);
        double r50321 = r50314 + r50320;
        double r50322 = 2.0;
        double r50323 = r50322 * r50308;
        double r50324 = r50321 / r50323;
        double r50325 = 6.2080577970805e+104;
        bool r50326 = r50302 <= r50325;
        double r50327 = 0.5;
        double r50328 = r50306 / r50327;
        double r50329 = r50314 - r50320;
        double r50330 = r50328 / r50329;
        double r50331 = -1.0;
        double r50332 = r50331 * r50307;
        double r50333 = r50326 ? r50330 : r50332;
        double r50334 = r50313 ? r50324 : r50333;
        double r50335 = r50304 ? r50311 : r50334;
        return r50335;
}

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 4 regimes

  2. if b < -8.301687926884189e+98

    1. Initial program 46.2

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -8.301687926884189e+98 < b < -7.892835993842436e-230

    1. Initial program 8.1

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

    if -7.892835993842436e-230 < b < 6.2080577970805e+104

    1. Initial program 29.9

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

    3. Applied flip-+30.0

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

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(c \cdot 4\right) \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num15.9

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + \left(c \cdot 4\right) \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified15.3

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\left(c \cdot 4\right) \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

    8. Taylor expanded around 0 9.8

      \[\leadsto \frac{1}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    9. Using strategy rm

    10. Applied associate-/r*9.5

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{0.5}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

    11. Simplified9.4

      \[\leadsto \frac{\color{blue}{\frac{c}{0.5}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

    if 6.2080577970805e+104 < b

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.301687926884188663878043402578250574713 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -7.89283599384243582562951448457087501988 \cdot 10^{-230}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.208057797080499790536355473922963434123 \cdot 10^{104}:\\ \;\;\;\;\frac{\frac{c}{0.5}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))