Average Error: 34.4 → 6.8
Time: 11.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5688227236985301 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.119187438943242 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 6.74838527698993 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.5688227236985301 \cdot 10^{105}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.119187438943242 \cdot 10^{-255}:\\
\;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

\mathbf{elif}\;b_2 \le 6.74838527698993 \cdot 10^{90}:\\
\;\;\;\;\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 r70659 = b_2;
        double r70660 = -r70659;
        double r70661 = r70659 * r70659;
        double r70662 = a;
        double r70663 = c;
        double r70664 = r70662 * r70663;
        double r70665 = r70661 - r70664;
        double r70666 = sqrt(r70665);
        double r70667 = r70660 - r70666;
        double r70668 = r70667 / r70662;
        return r70668;
}
↓
double f(double a, double b_2, double c) {
        double r70669 = b_2;
        double r70670 = -1.56882272369853e+105;
        bool r70671 = r70669 <= r70670;
        double r70672 = -0.5;
        double r70673 = c;
        double r70674 = r70673 / r70669;
        double r70675 = r70672 * r70674;
        double r70676 = 3.119187438943242e-255;
        bool r70677 = r70669 <= r70676;
        double r70678 = 1.0;
        double r70679 = r70678 / r70673;
        double r70680 = r70669 * r70669;
        double r70681 = a;
        double r70682 = r70681 * r70673;
        double r70683 = r70680 - r70682;
        double r70684 = sqrt(r70683);
        double r70685 = r70684 - r70669;
        double r70686 = r70679 * r70685;
        double r70687 = r70678 / r70686;
        double r70688 = 6.74838527698993e+90;
        bool r70689 = r70669 <= r70688;
        double r70690 = -r70669;
        double r70691 = r70690 - r70684;
        double r70692 = r70681 / r70691;
        double r70693 = r70678 / r70692;
        double r70694 = -2.0;
        double r70695 = r70669 / r70681;
        double r70696 = r70694 * r70695;
        double r70697 = r70689 ? r70693 : r70696;
        double r70698 = r70677 ? r70687 : r70697;
        double r70699 = r70671 ? r70675 : r70698;
        return r70699;
}

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 < -1.56882272369853e+105

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 2.5

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

    if -1.56882272369853e+105 < b_2 < 3.119187438943242e-255

    1. Initial program 31.0

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

    3. Applied flip--31.1

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied clear-num16.3

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

    8. Simplified15.6

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

    10. Applied clear-num15.6

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

    11. Simplified9.6

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

    if 3.119187438943242e-255 < b_2 < 6.74838527698993e+90

    1. Initial program 8.2

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

    3. Applied clear-num8.4

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

    if 6.74838527698993e+90 < b_2

    1. Initial program 45.6

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

    3. Applied flip--62.7

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    6. Taylor expanded around 0 4.6

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5688227236985301 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.119187438943242 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 6.74838527698993 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))