Average Error: 32.9 → 11.0
Time: 29.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 30831695.97060173:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.515406138267436 \cdot 10^{+130}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 30831695.97060173:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3073731 = b_2;
        double r3073732 = -r3073731;
        double r3073733 = r3073731 * r3073731;
        double r3073734 = a;
        double r3073735 = c;
        double r3073736 = r3073734 * r3073735;
        double r3073737 = r3073733 - r3073736;
        double r3073738 = sqrt(r3073737);
        double r3073739 = r3073732 - r3073738;
        double r3073740 = r3073739 / r3073734;
        return r3073740;
}
↓
double f(double a, double b_2, double c) {
        double r3073741 = b_2;
        double r3073742 = -1.515406138267436e+130;
        bool r3073743 = r3073741 <= r3073742;
        double r3073744 = -0.5;
        double r3073745 = c;
        double r3073746 = r3073745 / r3073741;
        double r3073747 = r3073744 * r3073746;
        double r3073748 = 30831695.97060173;
        bool r3073749 = r3073741 <= r3073748;
        double r3073750 = r3073741 * r3073741;
        double r3073751 = a;
        double r3073752 = r3073751 * r3073745;
        double r3073753 = r3073750 - r3073752;
        double r3073754 = sqrt(r3073753);
        double r3073755 = -r3073741;
        double r3073756 = r3073754 + r3073755;
        double r3073757 = r3073745 / r3073756;
        double r3073758 = 0.5;
        double r3073759 = r3073746 * r3073758;
        double r3073760 = 2.0;
        double r3073761 = r3073741 / r3073751;
        double r3073762 = r3073760 * r3073761;
        double r3073763 = r3073759 - r3073762;
        double r3073764 = r3073749 ? r3073757 : r3073763;
        double r3073765 = r3073743 ? r3073747 : r3073764;
        return r3073765;
}

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 3 regimes

  2. if b_2 < -1.515406138267436e+130

    1. Initial program 60.2

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

    2. Taylor expanded around inf 60.2

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified60.2

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

    5. Applied div-inv60.2

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

    6. Taylor expanded around -inf 2.3

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

    if -1.515406138267436e+130 < b_2 < 30831695.97060173

    1. Initial program 24.4

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

    2. Taylor expanded around inf 24.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified24.4

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

    5. Applied div-inv24.5

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

    7. Applied flip--30.9

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

    8. Applied associate-*l/31.0

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

    9. Simplified19.6

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

    10. Taylor expanded around inf 15.7

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

    if 30831695.97060173 < b_2

    1. Initial program 30.5

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

    2. Taylor expanded around inf 30.5

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified30.5

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

    4. Taylor expanded around inf 7.0

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.515406138267436 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 30831695.97060173:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))