Average Error: 33.9 → 9.1
Time: 5.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.7601787122462339 \cdot 10^{144}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -9.0165495816306676 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.2458828287346784 \cdot 10^{132}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.7601787122462339 \cdot 10^{144}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -9.0165495816306676 \cdot 10^{-123}:\\
\;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.2458828287346784 \cdot 10^{132}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r83652 = b_2;
        double r83653 = -r83652;
        double r83654 = r83652 * r83652;
        double r83655 = a;
        double r83656 = c;
        double r83657 = r83655 * r83656;
        double r83658 = r83654 - r83657;
        double r83659 = sqrt(r83658);
        double r83660 = r83653 - r83659;
        double r83661 = r83660 / r83655;
        return r83661;
}
↓
double f(double a, double b_2, double c) {
        double r83662 = b_2;
        double r83663 = -2.760178712246234e+144;
        bool r83664 = r83662 <= r83663;
        double r83665 = -0.5;
        double r83666 = c;
        double r83667 = r83666 / r83662;
        double r83668 = r83665 * r83667;
        double r83669 = -9.016549581630668e-123;
        bool r83670 = r83662 <= r83669;
        double r83671 = 1.0;
        double r83672 = a;
        double r83673 = r83672 * r83666;
        double r83674 = r83672 / r83673;
        double r83675 = r83662 * r83662;
        double r83676 = r83675 - r83673;
        double r83677 = sqrt(r83676);
        double r83678 = r83677 - r83662;
        double r83679 = r83674 * r83678;
        double r83680 = r83671 / r83679;
        double r83681 = 1.2458828287346784e+132;
        bool r83682 = r83662 <= r83681;
        double r83683 = -r83662;
        double r83684 = r83683 / r83672;
        double r83685 = r83677 / r83672;
        double r83686 = r83684 - r83685;
        double r83687 = -2.0;
        double r83688 = r83662 / r83672;
        double r83689 = r83687 * r83688;
        double r83690 = r83682 ? r83686 : r83689;
        double r83691 = r83670 ? r83680 : r83690;
        double r83692 = r83664 ? r83668 : r83691;
        return r83692;
}

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 < -2.760178712246234e+144

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 1.4

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

    if -2.760178712246234e+144 < b_2 < -9.016549581630668e-123

    1. Initial program 41.2

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

    3. Applied flip--41.2

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

    4. Simplified16.1

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

    5. Simplified16.1

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied clear-num16.4

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

    8. Simplified14.2

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

    if -9.016549581630668e-123 < b_2 < 1.2458828287346784e+132

    1. Initial program 11.5

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

    3. Applied div-sub11.5

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

    if 1.2458828287346784e+132 < b_2

    1. Initial program 56.4

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

    3. Applied flip--63.9

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

    4. Simplified62.8

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

    5. Simplified62.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    6. Taylor expanded around 0 3.2

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.7601787122462339 \cdot 10^{144}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -9.0165495816306676 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.2458828287346784 \cdot 10^{132}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))