Average Error: 34.6 → 9.4
Time: 18.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\
\;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r28765 = b_2;
        double r28766 = -r28765;
        double r28767 = r28765 * r28765;
        double r28768 = a;
        double r28769 = c;
        double r28770 = r28768 * r28769;
        double r28771 = r28767 - r28770;
        double r28772 = sqrt(r28771);
        double r28773 = r28766 - r28772;
        double r28774 = r28773 / r28768;
        return r28774;
}
↓
double f(double a, double b_2, double c) {
        double r28775 = b_2;
        double r28776 = -1.2929563087095085e+102;
        bool r28777 = r28775 <= r28776;
        double r28778 = -0.5;
        double r28779 = c;
        double r28780 = r28779 / r28775;
        double r28781 = r28778 * r28780;
        double r28782 = -1.8936269564254704e-242;
        bool r28783 = r28775 <= r28782;
        double r28784 = 1.0;
        double r28785 = r28775 * r28775;
        double r28786 = a;
        double r28787 = r28786 * r28779;
        double r28788 = r28785 - r28787;
        double r28789 = sqrt(r28788);
        double r28790 = r28789 - r28775;
        double r28791 = r28790 / r28786;
        double r28792 = r28791 / r28779;
        double r28793 = r28784 / r28792;
        double r28794 = r28784 / r28786;
        double r28795 = r28793 * r28794;
        double r28796 = 0.010648423176581222;
        bool r28797 = r28775 <= r28796;
        double r28798 = -r28775;
        double r28799 = r28798 - r28789;
        double r28800 = r28799 / r28786;
        double r28801 = 0.5;
        double r28802 = r28801 * r28780;
        double r28803 = 2.0;
        double r28804 = r28775 / r28786;
        double r28805 = r28803 * r28804;
        double r28806 = r28802 - r28805;
        double r28807 = r28797 ? r28800 : r28806;
        double r28808 = r28783 ? r28795 : r28807;
        double r28809 = r28777 ? r28781 : r28808;
        return r28809;
}

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 < -1.2929563087095085e+102

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 2.5

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

    if -1.2929563087095085e+102 < b_2 < -1.8936269564254704e-242

    1. Initial program 34.8

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

    3. Applied flip--34.9

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied div-inv16.6

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}}\]
    8. Using strategy rm

    9. Applied clear-num16.9

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

    10. Simplified16.9

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

    12. Applied associate-/r*15.1

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

    if -1.8936269564254704e-242 < b_2 < 0.010648423176581222

    1. Initial program 11.4

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

    if 0.010648423176581222 < b_2

    1. Initial program 32.9

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))