Average Error: 34.7 → 7.0
Time: 6.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.3202538172935113 \cdot 10^{68}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.67086091268017442 \cdot 10^{125}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r55255 = b;
        double r55256 = -r55255;
        double r55257 = r55255 * r55255;
        double r55258 = 4.0;
        double r55259 = a;
        double r55260 = r55258 * r55259;
        double r55261 = c;
        double r55262 = r55260 * r55261;
        double r55263 = r55257 - r55262;
        double r55264 = sqrt(r55263);
        double r55265 = r55256 + r55264;
        double r55266 = 2.0;
        double r55267 = r55266 * r55259;
        double r55268 = r55265 / r55267;
        return r55268;
}
↓
double f(double a, double b, double c) {
        double r55269 = b;
        double r55270 = -2.3202538172935113e+68;
        bool r55271 = r55269 <= r55270;
        double r55272 = 1.0;
        double r55273 = c;
        double r55274 = r55273 / r55269;
        double r55275 = a;
        double r55276 = r55269 / r55275;
        double r55277 = r55274 - r55276;
        double r55278 = r55272 * r55277;
        double r55279 = -8.908835082502404e-161;
        bool r55280 = r55269 <= r55279;
        double r55281 = -r55269;
        double r55282 = r55269 * r55269;
        double r55283 = 4.0;
        double r55284 = r55283 * r55275;
        double r55285 = r55284 * r55273;
        double r55286 = r55282 - r55285;
        double r55287 = sqrt(r55286);
        double r55288 = r55281 + r55287;
        double r55289 = 1.0;
        double r55290 = 2.0;
        double r55291 = r55290 * r55275;
        double r55292 = r55289 / r55291;
        double r55293 = r55288 * r55292;
        double r55294 = 3.6708609126801744e+125;
        bool r55295 = r55269 <= r55294;
        double r55296 = r55281 - r55287;
        double r55297 = r55296 / r55283;
        double r55298 = r55289 / r55297;
        double r55299 = r55298 / r55290;
        double r55300 = r55289 / r55273;
        double r55301 = r55289 / r55300;
        double r55302 = r55299 * r55301;
        double r55303 = -1.0;
        double r55304 = r55303 * r55274;
        double r55305 = r55295 ? r55302 : r55304;
        double r55306 = r55280 ? r55293 : r55305;
        double r55307 = r55271 ? r55278 : r55306;
        return r55307;
}

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 < -2.3202538172935113e+68

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 5.1

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

    3. Simplified5.1

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

    if -2.3202538172935113e+68 < b < -8.908835082502404e-161

    1. Initial program 6.3

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

    3. Applied div-inv6.5

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

    if -8.908835082502404e-161 < b < 3.6708609126801744e+125

    1. Initial program 29.8

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

    3. Applied flip-+30.1

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

    4. Simplified16.7

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

    6. Applied clear-num16.9

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

    7. Simplified16.9

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

    9. Applied div-inv17.5

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

    10. Applied add-sqr-sqrt17.5

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

    11. Applied times-frac17.3

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

    12. Applied times-frac16.4

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

    13. Simplified16.4

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

    14. Simplified15.7

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

    16. Applied clear-num15.8

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

    17. Simplified10.6

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

    if 3.6708609126801744e+125 < b

    1. Initial program 61.6

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))