Average Error: 33.7 → 9.0
Time: 5.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.0203295214211515 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 42194588.455395833:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\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 8.0203295214211515 \cdot 10^{-175}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{elif}\;b_2 \le 42194588.455395833:\\
\;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17543 = b_2;
        double r17544 = -r17543;
        double r17545 = r17543 * r17543;
        double r17546 = a;
        double r17547 = c;
        double r17548 = r17546 * r17547;
        double r17549 = r17545 - r17548;
        double r17550 = sqrt(r17549);
        double r17551 = r17544 + r17550;
        double r17552 = r17551 / r17546;
        return r17552;
}
↓
double f(double a, double b_2, double c) {
        double r17553 = b_2;
        double r17554 = -7.700313305414632e+138;
        bool r17555 = r17553 <= r17554;
        double r17556 = 0.5;
        double r17557 = c;
        double r17558 = r17557 / r17553;
        double r17559 = r17556 * r17558;
        double r17560 = 2.0;
        double r17561 = a;
        double r17562 = r17553 / r17561;
        double r17563 = r17560 * r17562;
        double r17564 = r17559 - r17563;
        double r17565 = 8.020329521421151e-175;
        bool r17566 = r17553 <= r17565;
        double r17567 = r17553 * r17553;
        double r17568 = r17561 * r17557;
        double r17569 = r17567 - r17568;
        double r17570 = sqrt(r17569);
        double r17571 = r17570 - r17553;
        double r17572 = r17571 / r17561;
        double r17573 = 42194588.45539583;
        bool r17574 = r17553 <= r17573;
        double r17575 = 0.0;
        double r17576 = r17575 + r17568;
        double r17577 = -r17553;
        double r17578 = r17577 - r17570;
        double r17579 = r17576 / r17578;
        double r17580 = r17579 / r17561;
        double r17581 = -0.5;
        double r17582 = r17581 * r17558;
        double r17583 = r17574 ? r17580 : r17582;
        double r17584 = r17566 ? r17572 : r17583;
        double r17585 = r17555 ? r17564 : r17584;
        return r17585;
}

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 < 8.020329521421151e-175

    1. Initial program 10.4

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

    3. Applied *-un-lft-identity10.4

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

    4. Applied associate-/r*10.4

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

    5. Simplified10.4

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

    if 8.020329521421151e-175 < b_2 < 42194588.45539583

    1. Initial program 31.5

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

    3. Applied flip-+31.5

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

    4. Simplified17.1

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

    if 42194588.45539583 < 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 simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.70031330541463201 \cdot 10^{138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 8.0203295214211515 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \le 42194588.455395833:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))