Average Error: 34.5 → 6.6
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\
\;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r25642 = b_2;
        double r25643 = -r25642;
        double r25644 = r25642 * r25642;
        double r25645 = a;
        double r25646 = c;
        double r25647 = r25645 * r25646;
        double r25648 = r25644 - r25647;
        double r25649 = sqrt(r25648);
        double r25650 = r25643 - r25649;
        double r25651 = r25650 / r25645;
        return r25651;
}
↓
double f(double a, double b_2, double c) {
        double r25652 = b_2;
        double r25653 = -5.602441184943773e+118;
        bool r25654 = r25652 <= r25653;
        double r25655 = -0.5;
        double r25656 = c;
        double r25657 = r25656 / r25652;
        double r25658 = r25655 * r25657;
        double r25659 = -3.897979690537011e-281;
        bool r25660 = r25652 <= r25659;
        double r25661 = 2.0;
        double r25662 = pow(r25652, r25661);
        double r25663 = a;
        double r25664 = r25663 * r25656;
        double r25665 = r25662 - r25664;
        double r25666 = sqrt(r25665);
        double r25667 = r25666 - r25652;
        double r25668 = r25656 / r25667;
        double r25669 = 2.1255630798514387e+135;
        bool r25670 = r25652 <= r25669;
        double r25671 = -r25652;
        double r25672 = r25652 * r25652;
        double r25673 = r25672 - r25664;
        double r25674 = sqrt(r25673);
        double r25675 = r25671 - r25674;
        double r25676 = r25675 / r25663;
        double r25677 = 0.5;
        double r25678 = r25677 * r25657;
        double r25679 = r25652 / r25663;
        double r25680 = r25661 * r25679;
        double r25681 = r25678 - r25680;
        double r25682 = r25670 ? r25676 : r25681;
        double r25683 = r25660 ? r25668 : r25682;
        double r25684 = r25654 ? r25658 : r25683;
        return r25684;
}

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 < -5.602441184943773e+118

    1. Initial program 61.1

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

    2. Taylor expanded around -inf 1.9

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

    if -5.602441184943773e+118 < b_2 < -3.897979690537011e-281

    1. Initial program 34.7

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

    3. Applied flip--34.7

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

    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-frac15.0

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

    9. Applied associate-/l*10.5

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

    10. Simplified8.4

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

    11. Taylor expanded around 0 8.4

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

    if -3.897979690537011e-281 < b_2 < 2.1255630798514387e+135

    1. Initial program 9.3

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

    if 2.1255630798514387e+135 < b_2

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.602441184943772642330945248646923860899 \cdot 10^{118}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.897979690537010916247637791104885449418 \cdot 10^{-281}:\\ \;\;\;\;\frac{c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))