Average Error: 33.6 → 9.9
Time: 15.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.97357776541463793 \cdot 10^{146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.01021101907822424 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.97357776541463793 \cdot 10^{146}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 5.01021101907822424 \cdot 10^{-85}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r22585 = b_2;
        double r22586 = -r22585;
        double r22587 = r22585 * r22585;
        double r22588 = a;
        double r22589 = c;
        double r22590 = r22588 * r22589;
        double r22591 = r22587 - r22590;
        double r22592 = sqrt(r22591);
        double r22593 = r22586 + r22592;
        double r22594 = r22593 / r22588;
        return r22594;
}
↓
double f(double a, double b_2, double c) {
        double r22595 = b_2;
        double r22596 = -2.973577765414638e+146;
        bool r22597 = r22595 <= r22596;
        double r22598 = 0.5;
        double r22599 = c;
        double r22600 = r22599 / r22595;
        double r22601 = r22598 * r22600;
        double r22602 = 2.0;
        double r22603 = a;
        double r22604 = r22595 / r22603;
        double r22605 = r22602 * r22604;
        double r22606 = r22601 - r22605;
        double r22607 = 5.010211019078224e-85;
        bool r22608 = r22595 <= r22607;
        double r22609 = r22595 * r22595;
        double r22610 = r22603 * r22599;
        double r22611 = r22609 - r22610;
        double r22612 = sqrt(r22611);
        double r22613 = r22612 - r22595;
        double r22614 = r22613 / r22603;
        double r22615 = -0.5;
        double r22616 = r22615 * r22600;
        double r22617 = r22608 ? r22614 : r22616;
        double r22618 = r22597 ? r22606 : r22617;
        return r22618;
}

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 < -2.973577765414638e+146

    1. Initial program 60.6

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

    2. Simplified60.6

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

    3. Taylor expanded around -inf 2.2

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

    if -2.973577765414638e+146 < b_2 < 5.010211019078224e-85

    1. Initial program 11.5

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

    2. Simplified11.5

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

    4. Applied clear-num11.6

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

    6. Applied div-inv11.7

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

    7. Applied add-cube-cbrt11.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    8. Applied times-frac11.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]

    9. Simplified11.6

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    10. Simplified11.6

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

    12. Applied associate-*l/11.5

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

    13. Simplified11.5

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

    if 5.010211019078224e-85 < b_2

    1. Initial program 52.4

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

    2. Simplified52.4

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

    3. Taylor expanded around inf 10.2

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.97357776541463793 \cdot 10^{146}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 5.01021101907822424 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))