Average Error: 32.8 → 9.8
Time: 22.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r402577 = b_2;
        double r402578 = -r402577;
        double r402579 = r402577 * r402577;
        double r402580 = a;
        double r402581 = c;
        double r402582 = r402580 * r402581;
        double r402583 = r402579 - r402582;
        double r402584 = sqrt(r402583);
        double r402585 = r402578 + r402584;
        double r402586 = r402585 / r402580;
        return r402586;
}
↓
double f(double a, double b_2, double c) {
        double r402587 = b_2;
        double r402588 = -1.3599228730895225e+90;
        bool r402589 = r402587 <= r402588;
        double r402590 = 0.5;
        double r402591 = c;
        double r402592 = r402591 / r402587;
        double r402593 = r402590 * r402592;
        double r402594 = a;
        double r402595 = r402587 / r402594;
        double r402596 = r402593 - r402595;
        double r402597 = r402596 - r402595;
        double r402598 = 3.1295384133612364e-73;
        bool r402599 = r402587 <= r402598;
        double r402600 = r402587 * r402587;
        double r402601 = r402594 * r402591;
        double r402602 = r402600 - r402601;
        double r402603 = sqrt(r402602);
        double r402604 = r402603 / r402594;
        double r402605 = r402604 - r402595;
        double r402606 = -0.5;
        double r402607 = r402592 * r402606;
        double r402608 = r402599 ? r402605 : r402607;
        double r402609 = r402589 ? r402597 : r402608;
        return r402609;
}

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 3 regimes

  2. if b_2 < -1.3599228730895225e+90

    1. Initial program 41.6

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

    2. Simplified41.6

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-sub41.6

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

    5. Taylor expanded around -inf 3.8

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

    if -1.3599228730895225e+90 < b_2 < 3.1295384133612364e-73

    1. Initial program 12.8

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

    2. Simplified12.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-sub12.8

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]
    5. Using strategy rm

    6. Applied clear-num12.9

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}} - \frac{b_2}{a}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity12.9

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

    9. Applied *-un-lft-identity12.9

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

    10. Applied distribute-lft-out--12.9

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

    11. Simplified12.8

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

    if 3.1295384133612364e-73 < b_2

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.0

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.3599228730895225 \cdot 10^{+90}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))