Average Error: 34.7 → 6.7
Time: 6.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\
\;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r18442 = b_2;
        double r18443 = -r18442;
        double r18444 = r18442 * r18442;
        double r18445 = a;
        double r18446 = c;
        double r18447 = r18445 * r18446;
        double r18448 = r18444 - r18447;
        double r18449 = sqrt(r18448);
        double r18450 = r18443 + r18449;
        double r18451 = r18450 / r18445;
        return r18451;
}
↓
double f(double a, double b_2, double c) {
        double r18452 = b_2;
        double r18453 = -1.9721792334777768e+151;
        bool r18454 = r18452 <= r18453;
        double r18455 = 0.5;
        double r18456 = c;
        double r18457 = r18456 / r18452;
        double r18458 = r18455 * r18457;
        double r18459 = 2.0;
        double r18460 = a;
        double r18461 = r18452 / r18460;
        double r18462 = r18459 * r18461;
        double r18463 = r18458 - r18462;
        double r18464 = 7.484772368654571e-258;
        bool r18465 = r18452 <= r18464;
        double r18466 = -r18452;
        double r18467 = r18452 * r18452;
        double r18468 = r18460 * r18456;
        double r18469 = r18467 - r18468;
        double r18470 = sqrt(r18469);
        double r18471 = r18466 + r18470;
        double r18472 = r18471 / r18460;
        double r18473 = 9.792086348653723e+126;
        bool r18474 = r18452 <= r18473;
        double r18475 = r18466 - r18470;
        double r18476 = r18456 / r18475;
        double r18477 = -0.5;
        double r18478 = r18477 * r18457;
        double r18479 = r18474 ? r18476 : r18478;
        double r18480 = r18465 ? r18472 : r18479;
        double r18481 = r18454 ? r18463 : r18480;
        return r18481;
}

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 < -1.9721792334777768e+151

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.9

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

    if -1.9721792334777768e+151 < b_2 < 7.484772368654571e-258

    1. Initial program 9.8

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

    if 7.484772368654571e-258 < b_2 < 9.792086348653723e+126

    1. Initial program 34.4

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

    3. Applied flip-+34.5

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

    4. Simplified16.0

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

    6. Applied clear-num16.1

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

    7. Simplified14.8

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

    9. Applied associate-/r*14.6

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

    10. Simplified8.1

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

    if 9.792086348653723e+126 < b_2

    1. Initial program 61.4

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

    2. Taylor expanded around inf 2.1

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.9721792334777768 \cdot 10^{151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.48477236865457127 \cdot 10^{-258}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{elif}\;b_2 \le 9.79208634865372271 \cdot 10^{126}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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))