Average Error: 34.4 → 7.5
Time: 8.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.896155291582558 \cdot 10^{-285}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 2445759453.4737968:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0461303908572575 \cdot 10^{65}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -3.896155291582558 \cdot 10^{-285}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 2445759453.4737968:\\
\;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r29177 = b_2;
        double r29178 = -r29177;
        double r29179 = r29177 * r29177;
        double r29180 = a;
        double r29181 = c;
        double r29182 = r29180 * r29181;
        double r29183 = r29179 - r29182;
        double r29184 = sqrt(r29183);
        double r29185 = r29178 + r29184;
        double r29186 = r29185 / r29180;
        return r29186;
}
↓
double f(double a, double b_2, double c) {
        double r29187 = b_2;
        double r29188 = -1.0461303908572575e+65;
        bool r29189 = r29187 <= r29188;
        double r29190 = 0.5;
        double r29191 = c;
        double r29192 = r29191 / r29187;
        double r29193 = r29190 * r29192;
        double r29194 = 2.0;
        double r29195 = a;
        double r29196 = r29187 / r29195;
        double r29197 = r29194 * r29196;
        double r29198 = r29193 - r29197;
        double r29199 = -3.896155291582558e-285;
        bool r29200 = r29187 <= r29199;
        double r29201 = -r29187;
        double r29202 = r29187 * r29187;
        double r29203 = r29195 * r29191;
        double r29204 = r29202 - r29203;
        double r29205 = sqrt(r29204);
        double r29206 = r29201 + r29205;
        double r29207 = 1.0;
        double r29208 = r29207 / r29195;
        double r29209 = r29206 * r29208;
        double r29210 = 2445759453.473797;
        bool r29211 = r29187 <= r29210;
        double r29212 = r29201 - r29205;
        double r29213 = r29212 / r29191;
        double r29214 = r29207 / r29213;
        double r29215 = -0.5;
        double r29216 = r29215 * r29192;
        double r29217 = r29211 ? r29214 : r29216;
        double r29218 = r29200 ? r29209 : r29217;
        double r29219 = r29189 ? r29198 : r29218;
        return r29219;
}

Error

Bits error versus a

Bits error versus b_2

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 < -1.0461303908572575e+65

    1. Initial program 41.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 4.6

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -1.0461303908572575e+65 < b_2 < -3.896155291582558e-285

    1. Initial program 9.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv10.0

      \[\leadsto \color{blue}{\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if -3.896155291582558e-285 < b_2 < 2445759453.473797

    1. Initial program 25.6

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+25.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified17.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

    7. Applied associate-/r*17.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{1}}{a}}\]

    8. Simplified14.3

      \[\leadsto \frac{\color{blue}{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}{a}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity14.3

      \[\leadsto \frac{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{\color{blue}{1 \cdot c}}}}{a}\]

    11. Applied *-un-lft-identity14.3

      \[\leadsto \frac{\frac{a}{\frac{\color{blue}{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{1 \cdot c}}}{a}\]

    12. Applied times-frac14.3

      \[\leadsto \frac{\frac{a}{\color{blue}{\frac{1}{1} \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}{a}\]

    13. Applied *-un-lft-identity14.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot a}}{\frac{1}{1} \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\]

    14. Applied times-frac14.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{1}{1}} \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}{a}\]

    15. Applied associate-/l*14.2

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{1}{1}}}{\frac{a}{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}}\]

    16. Simplified10.8

      \[\leadsto \frac{\frac{1}{\frac{1}{1}}}{\color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}\]

    if 2445759453.473797 < b_2

    1. Initial program 56.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.9

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0461303908572575 \cdot 10^{65}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.896155291582558 \cdot 10^{-285}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 2445759453.4737968:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))