Average Error: 34.0 → 6.5
Time: 15.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.077239628548905378565124203356585166888 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.657998399682531790024702628233707002655 \cdot 10^{-302}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 3.361636799726259566012474460441175928147 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.077239628548905378565124203356585166888 \cdot 10^{154}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.657998399682531790024702628233707002655 \cdot 10^{-302}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 3.361636799726259566012474460441175928147 \cdot 10^{93}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r78660 = b_2;
        double r78661 = -r78660;
        double r78662 = r78660 * r78660;
        double r78663 = a;
        double r78664 = c;
        double r78665 = r78663 * r78664;
        double r78666 = r78662 - r78665;
        double r78667 = sqrt(r78666);
        double r78668 = r78661 - r78667;
        double r78669 = r78668 / r78663;
        return r78669;
}
↓
double f(double a, double b_2, double c) {
        double r78670 = b_2;
        double r78671 = -1.0772396285489054e+154;
        bool r78672 = r78670 <= r78671;
        double r78673 = -0.5;
        double r78674 = c;
        double r78675 = r78674 / r78670;
        double r78676 = r78673 * r78675;
        double r78677 = 3.657998399682532e-302;
        bool r78678 = r78670 <= r78677;
        double r78679 = r78670 * r78670;
        double r78680 = a;
        double r78681 = r78680 * r78674;
        double r78682 = r78679 - r78681;
        double r78683 = sqrt(r78682);
        double r78684 = r78683 - r78670;
        double r78685 = r78674 / r78684;
        double r78686 = 3.3616367997262596e+93;
        bool r78687 = r78670 <= r78686;
        double r78688 = 1.0;
        double r78689 = -r78670;
        double r78690 = r78689 - r78683;
        double r78691 = r78680 / r78690;
        double r78692 = r78688 / r78691;
        double r78693 = -2.0;
        double r78694 = r78670 / r78680;
        double r78695 = r78693 * r78694;
        double r78696 = r78687 ? r78692 : r78695;
        double r78697 = r78678 ? r78685 : r78696;
        double r78698 = r78672 ? r78676 : r78697;
        return r78698;
}

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 < -1.0772396285489054e+154

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.8

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

    if -1.0772396285489054e+154 < b_2 < 3.657998399682532e-302

    1. Initial program 34.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    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. Simplified15.9

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

    5. Simplified15.9

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

    7. Applied *-un-lft-identity15.9

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

    8. Applied times-frac15.1

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

    9. Applied associate-/l*10.7

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

    10. Simplified8.1

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

    if 3.657998399682532e-302 < b_2 < 3.3616367997262596e+93

    1. Initial program 8.9

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

    3. Applied clear-num9.1

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

    if 3.3616367997262596e+93 < b_2

    1. Initial program 45.2

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

    3. Applied flip--62.9

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

    4. Simplified62.0

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

    5. Simplified62.0

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

    6. Taylor expanded around 0 3.5

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.077239628548905378565124203356585166888 \cdot 10^{154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.657998399682531790024702628233707002655 \cdot 10^{-302}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 3.361636799726259566012474460441175928147 \cdot 10^{93}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))