Average Error: 33.9 → 6.7
Time: 21.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r29637 = b_2;
        double r29638 = -r29637;
        double r29639 = r29637 * r29637;
        double r29640 = a;
        double r29641 = c;
        double r29642 = r29640 * r29641;
        double r29643 = r29639 - r29642;
        double r29644 = sqrt(r29643);
        double r29645 = r29638 - r29644;
        double r29646 = r29645 / r29640;
        return r29646;
}
↓
double f(double a, double b_2, double c) {
        double r29647 = b_2;
        double r29648 = -5.674469085146397e+110;
        bool r29649 = r29647 <= r29648;
        double r29650 = -0.5;
        double r29651 = c;
        double r29652 = r29651 / r29647;
        double r29653 = r29650 * r29652;
        double r29654 = 1.4099136946117732e-265;
        bool r29655 = r29647 <= r29654;
        double r29656 = r29647 * r29647;
        double r29657 = a;
        double r29658 = r29657 * r29651;
        double r29659 = r29656 - r29658;
        double r29660 = sqrt(r29659);
        double r29661 = r29660 - r29647;
        double r29662 = r29651 / r29661;
        double r29663 = 1.7151811081882383e+78;
        bool r29664 = r29647 <= r29663;
        double r29665 = -r29647;
        double r29666 = r29665 - r29660;
        double r29667 = 1.0;
        double r29668 = r29667 / r29657;
        double r29669 = r29666 * r29668;
        double r29670 = -2.0;
        double r29671 = r29670 * r29647;
        double r29672 = r29671 / r29657;
        double r29673 = r29664 ? r29669 : r29672;
        double r29674 = r29655 ? r29662 : r29673;
        double r29675 = r29649 ? r29653 : r29674;
        return r29675;
}

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 < -5.674469085146397e+110

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.7

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

    if -5.674469085146397e+110 < b_2 < 1.4099136946117732e-265

    1. Initial program 31.8

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

    3. Applied flip--31.8

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac14.9

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

    9. Applied associate-/l*10.4

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

    10. Simplified8.7

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

    if 1.4099136946117732e-265 < b_2 < 1.7151811081882383e+78

    1. Initial program 8.5

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

    3. Applied div-inv8.6

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

    if 1.7151811081882383e+78 < b_2

    1. Initial program 43.0

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

    3. Applied flip--62.6

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    6. Taylor expanded around 0 4.8

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))