Average Error: 33.7 → 8.2
Time: 5.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.82920021484925195 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 25449170.687741734:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -4.82920021484925195 \cdot 10^{-294}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 25449170.687741734:\\
\;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19590 = b_2;
        double r19591 = -r19590;
        double r19592 = r19590 * r19590;
        double r19593 = a;
        double r19594 = c;
        double r19595 = r19593 * r19594;
        double r19596 = r19592 - r19595;
        double r19597 = sqrt(r19596);
        double r19598 = r19591 + r19597;
        double r19599 = r19598 / r19593;
        return r19599;
}
↓
double f(double a, double b_2, double c) {
        double r19600 = b_2;
        double r19601 = -7.700313305414632e+138;
        bool r19602 = r19600 <= r19601;
        double r19603 = 0.5;
        double r19604 = c;
        double r19605 = r19604 / r19600;
        double r19606 = r19603 * r19605;
        double r19607 = 2.0;
        double r19608 = a;
        double r19609 = r19600 / r19608;
        double r19610 = r19607 * r19609;
        double r19611 = r19606 - r19610;
        double r19612 = -4.829200214849252e-294;
        bool r19613 = r19600 <= r19612;
        double r19614 = -r19600;
        double r19615 = r19600 * r19600;
        double r19616 = r19608 * r19604;
        double r19617 = r19615 - r19616;
        double r19618 = sqrt(r19617);
        double r19619 = r19614 + r19618;
        double r19620 = r19619 / r19608;
        double r19621 = 25449170.687741734;
        bool r19622 = r19600 <= r19621;
        double r19623 = 1.0;
        double r19624 = r19614 - r19618;
        double r19625 = r19624 / r19604;
        double r19626 = r19608 / r19625;
        double r19627 = r19623 * r19626;
        double r19628 = r19627 / r19608;
        double r19629 = -0.5;
        double r19630 = r19629 * r19605;
        double r19631 = r19622 ? r19628 : r19630;
        double r19632 = r19613 ? r19620 : r19631;
        double r19633 = r19602 ? r19611 : r19632;
        return r19633;
}

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 < -7.700313305414632e+138

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 2.9

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

    if -7.700313305414632e+138 < b_2 < -4.829200214849252e-294

    1. Initial program 8.7

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

    if -4.829200214849252e-294 < b_2 < 25449170.687741734

    1. Initial program 25.5

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

    3. Applied flip-+25.5

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

    4. Simplified17.0

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

    6. Applied *-un-lft-identity17.0

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

    7. Applied *-un-lft-identity17.0

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

    8. Applied times-frac17.0

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

    9. Simplified17.0

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

    10. Simplified13.8

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

    if 25449170.687741734 < b_2

    1. Initial program 55.9

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

    2. Taylor expanded around inf 5.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -4.82920021484925195 \cdot 10^{-294}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 25449170.687741734:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))