Average Error: 28.4 → 16.7
Time: 12.1s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 183.1843483503574816495529375970363616943:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \frac{a \cdot c}{b}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\\ \end{array}\]
\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 183.1843483503574816495529375970363616943:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(-2 \cdot \frac{a \cdot c}{b}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\\

\end{array}
double f(double a, double b, double c) {
        double r29177 = b;
        double r29178 = -r29177;
        double r29179 = r29177 * r29177;
        double r29180 = 4.0;
        double r29181 = a;
        double r29182 = r29180 * r29181;
        double r29183 = c;
        double r29184 = r29182 * r29183;
        double r29185 = r29179 - r29184;
        double r29186 = sqrt(r29185);
        double r29187 = r29178 + r29186;
        double r29188 = 2.0;
        double r29189 = r29188 * r29181;
        double r29190 = r29187 / r29189;
        return r29190;
}
↓
double f(double a, double b, double c) {
        double r29191 = b;
        double r29192 = 183.18434835035748;
        bool r29193 = r29191 <= r29192;
        double r29194 = r29191 * r29191;
        double r29195 = 4.0;
        double r29196 = a;
        double r29197 = r29195 * r29196;
        double r29198 = c;
        double r29199 = r29197 * r29198;
        double r29200 = r29194 - r29199;
        double r29201 = r29200 - r29194;
        double r29202 = sqrt(r29200);
        double r29203 = r29202 + r29191;
        double r29204 = r29201 / r29203;
        double r29205 = 2.0;
        double r29206 = r29205 * r29196;
        double r29207 = r29204 / r29206;
        double r29208 = -2.0;
        double r29209 = r29196 * r29198;
        double r29210 = r29209 / r29191;
        double r29211 = r29208 * r29210;
        double r29212 = 1.0;
        double r29213 = r29212 / r29206;
        double r29214 = sqrt(r29213);
        double r29215 = r29211 * r29214;
        double r29216 = r29215 * r29214;
        double r29217 = r29193 ? r29207 : r29216;
        return r29217;
}

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 < 183.18434835035748

    1. Initial program 15.7

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

    2. Simplified15.7

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

    4. Applied flip--15.7

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

    5. Simplified14.7

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

    if 183.18434835035748 < b

    1. Initial program 34.7

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

    2. Simplified34.7

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

    3. Taylor expanded around inf 17.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a}\]
    4. Using strategy rm

    5. Applied div-inv17.6

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

    7. Applied add-sqr-sqrt17.6

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

    8. Applied associate-*r*17.6

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \frac{a \cdot c}{b}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 183.1843483503574816495529375970363616943:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \frac{a \cdot c}{b}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\right) \cdot \sqrt{\frac{1}{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))