Average Error: 34.4 → 6.8
Time: 11.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5688227236985301 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.119187438943242 \cdot 10^{-255}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{elif}\;b_2 \le 6.74838527698993 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{-2 \cdot b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r70778 = b_2;
        double r70779 = -r70778;
        double r70780 = r70778 * r70778;
        double r70781 = a;
        double r70782 = c;
        double r70783 = r70781 * r70782;
        double r70784 = r70780 - r70783;
        double r70785 = sqrt(r70784);
        double r70786 = r70779 - r70785;
        double r70787 = r70786 / r70781;
        return r70787;
}
↓
double f(double a, double b_2, double c) {
        double r70788 = b_2;
        double r70789 = -1.56882272369853e+105;
        bool r70790 = r70788 <= r70789;
        double r70791 = -0.5;
        double r70792 = c;
        double r70793 = r70792 / r70788;
        double r70794 = r70791 * r70793;
        double r70795 = 3.119187438943242e-255;
        bool r70796 = r70788 <= r70795;
        double r70797 = 1.0;
        double r70798 = r70797 / r70792;
        double r70799 = r70788 * r70788;
        double r70800 = a;
        double r70801 = r70800 * r70792;
        double r70802 = r70799 - r70801;
        double r70803 = sqrt(r70802);
        double r70804 = r70803 - r70788;
        double r70805 = r70798 * r70804;
        double r70806 = r70797 / r70805;
        double r70807 = 6.74838527698993e+90;
        bool r70808 = r70788 <= r70807;
        double r70809 = -r70788;
        double r70810 = r70809 - r70803;
        double r70811 = r70800 / r70810;
        double r70812 = r70797 / r70811;
        double r70813 = -2.0;
        double r70814 = r70813 * r70788;
        double r70815 = r70814 / r70800;
        double r70816 = r70808 ? r70812 : r70815;
        double r70817 = r70796 ? r70806 : r70816;
        double r70818 = r70790 ? r70794 : r70817;
        return r70818;
}

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 *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Applied associate-/l*16.3

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

    11. Simplified15.6

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

    13. Applied clear-num15.6

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

    14. Simplified9.6

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

    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 4.7

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

  4. Final simplification6.8

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

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))