Average Error: 34.4 → 8.2
Time: 5.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -7.774817858016759874058349302107218860939 \cdot 10^{-301}:\\ \;\;\;\;\frac{1 \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{elif}\;b_2 \le 3.163227370458367280461441428752527041014 \cdot 10^{53}:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -7.774817858016759874058349302107218860939 \cdot 10^{-301}:\\
\;\;\;\;\frac{1 \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\mathbf{elif}\;b_2 \le 3.163227370458367280461441428752527041014 \cdot 10^{53}:\\
\;\;\;\;\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 r16565 = b_2;
        double r16566 = -r16565;
        double r16567 = r16565 * r16565;
        double r16568 = a;
        double r16569 = c;
        double r16570 = r16568 * r16569;
        double r16571 = r16567 - r16570;
        double r16572 = sqrt(r16571);
        double r16573 = r16566 + r16572;
        double r16574 = r16573 / r16568;
        return r16574;
}
↓
double f(double a, double b_2, double c) {
        double r16575 = b_2;
        double r16576 = -9.912439588753869e+101;
        bool r16577 = r16575 <= r16576;
        double r16578 = 0.5;
        double r16579 = c;
        double r16580 = r16579 / r16575;
        double r16581 = r16578 * r16580;
        double r16582 = 2.0;
        double r16583 = a;
        double r16584 = r16575 / r16583;
        double r16585 = r16582 * r16584;
        double r16586 = r16581 - r16585;
        double r16587 = -7.77481785801676e-301;
        bool r16588 = r16575 <= r16587;
        double r16589 = 1.0;
        double r16590 = -r16575;
        double r16591 = r16575 * r16575;
        double r16592 = r16583 * r16579;
        double r16593 = r16591 - r16592;
        double r16594 = sqrt(r16593);
        double r16595 = r16590 + r16594;
        double r16596 = r16589 * r16595;
        double r16597 = r16596 / r16583;
        double r16598 = 3.1632273704583673e+53;
        bool r16599 = r16575 <= r16598;
        double r16600 = r16590 - r16594;
        double r16601 = r16600 / r16579;
        double r16602 = r16583 / r16601;
        double r16603 = r16589 * r16602;
        double r16604 = r16603 / r16583;
        double r16605 = -0.5;
        double r16606 = r16605 * r16580;
        double r16607 = r16599 ? r16604 : r16606;
        double r16608 = r16588 ? r16597 : r16607;
        double r16609 = r16577 ? r16586 : r16608;
        return r16609;
}

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 < -9.912439588753869e+101

    1. Initial program 46.9

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

    2. Taylor expanded around -inf 3.7

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

    if -9.912439588753869e+101 < b_2 < -7.77481785801676e-301

    1. Initial program 9.1

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

    3. Applied *-un-lft-identity9.1

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

    if -7.77481785801676e-301 < b_2 < 3.1632273704583673e+53

    1. Initial program 30.2

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

    3. Applied flip-+30.2

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

    4. Simplified17.8

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

    6. Applied *-un-lft-identity17.8

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

    7. Applied *-un-lft-identity17.8

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

    8. Applied times-frac17.8

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

    9. Simplified17.8

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

    10. Simplified14.8

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

    if 3.1632273704583673e+53 < b_2

    1. Initial program 57.6

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.91243958875386880555748684589545292526 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -7.774817858016759874058349302107218860939 \cdot 10^{-301}:\\ \;\;\;\;\frac{1 \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{elif}\;b_2 \le 3.163227370458367280461441428752527041014 \cdot 10^{53}:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))