Average Error: 34.4 → 6.7
Time: 6.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.5038627888747993 \cdot 10^{151}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.10735029247724072 \cdot 10^{-236}:\\ \;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot c\\ \mathbf{elif}\;b_2 \le 5.43153376685683538 \cdot 10^{106}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.5038627888747993 \cdot 10^{151}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.10735029247724072 \cdot 10^{-236}:\\
\;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot c\\

\mathbf{elif}\;b_2 \le 5.43153376685683538 \cdot 10^{106}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r22653 = b_2;
        double r22654 = -r22653;
        double r22655 = r22653 * r22653;
        double r22656 = a;
        double r22657 = c;
        double r22658 = r22656 * r22657;
        double r22659 = r22655 - r22658;
        double r22660 = sqrt(r22659);
        double r22661 = r22654 - r22660;
        double r22662 = r22661 / r22656;
        return r22662;
}
↓
double f(double a, double b_2, double c) {
        double r22663 = b_2;
        double r22664 = -5.503862788874799e+151;
        bool r22665 = r22663 <= r22664;
        double r22666 = -0.5;
        double r22667 = c;
        double r22668 = r22667 / r22663;
        double r22669 = r22666 * r22668;
        double r22670 = 1.1073502924772407e-236;
        bool r22671 = r22663 <= r22670;
        double r22672 = 1.0;
        double r22673 = r22663 * r22663;
        double r22674 = a;
        double r22675 = r22674 * r22667;
        double r22676 = r22673 - r22675;
        double r22677 = sqrt(r22676);
        double r22678 = r22677 - r22663;
        double r22679 = r22672 / r22678;
        double r22680 = r22679 * r22667;
        double r22681 = 5.431533766856835e+106;
        bool r22682 = r22663 <= r22681;
        double r22683 = -r22663;
        double r22684 = r22683 - r22677;
        double r22685 = r22674 / r22684;
        double r22686 = r22672 / r22685;
        double r22687 = 0.5;
        double r22688 = r22687 * r22668;
        double r22689 = 2.0;
        double r22690 = r22663 / r22674;
        double r22691 = r22689 * r22690;
        double r22692 = r22688 - r22691;
        double r22693 = r22682 ? r22686 : r22692;
        double r22694 = r22671 ? r22680 : r22693;
        double r22695 = r22665 ? r22669 : r22694;
        return r22695;
}

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 < -5.503862788874799e+151

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 1.5

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

    if -5.503862788874799e+151 < b_2 < 1.1073502924772407e-236

    1. Initial program 32.5

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

    3. Applied flip--32.6

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

    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

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

    8. Applied associate-/r*16.2

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

    9. Simplified14.2

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

    11. Applied div-inv14.2

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

    12. Applied *-un-lft-identity14.2

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

    13. Applied times-frac16.2

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

    14. Applied associate-/l*14.9

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

    15. Simplified8.8

      \[\leadsto \frac{\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{\frac{1}{c}}}\]
    16. Using strategy rm

    17. Applied div-inv8.8

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

    18. Simplified8.8

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

    if 1.1073502924772407e-236 < b_2 < 5.431533766856835e+106

    1. Initial program 8.6

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

    3. Applied clear-num8.7

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

    if 5.431533766856835e+106 < b_2

    1. Initial program 50.0

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

    2. Taylor expanded around inf 3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.5038627888747993 \cdot 10^{151}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.10735029247724072 \cdot 10^{-236}:\\ \;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot c\\ \mathbf{elif}\;b_2 \le 5.43153376685683538 \cdot 10^{106}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))