Average Error: 34.8 → 6.9
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.30882648567117255 \cdot 10^{71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.1047243661018065 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.1848561515745898 \cdot 10^{127}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.30882648567117255 \cdot 10^{71}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -7.1047243661018065 \cdot 10^{-302}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot 0 + c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 6.1848561515745898 \cdot 10^{127}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{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 r14110 = b_2;
        double r14111 = -r14110;
        double r14112 = r14110 * r14110;
        double r14113 = a;
        double r14114 = c;
        double r14115 = r14113 * r14114;
        double r14116 = r14112 - r14115;
        double r14117 = sqrt(r14116);
        double r14118 = r14111 - r14117;
        double r14119 = r14118 / r14113;
        return r14119;
}
↓
double f(double a, double b_2, double c) {
        double r14120 = b_2;
        double r14121 = -2.3088264856711725e+71;
        bool r14122 = r14120 <= r14121;
        double r14123 = -0.5;
        double r14124 = c;
        double r14125 = r14124 / r14120;
        double r14126 = r14123 * r14125;
        double r14127 = -7.104724366101806e-302;
        bool r14128 = r14120 <= r14127;
        double r14129 = 1.0;
        double r14130 = a;
        double r14131 = r14129 / r14130;
        double r14132 = 0.0;
        double r14133 = r14131 * r14132;
        double r14134 = r14133 + r14124;
        double r14135 = -r14120;
        double r14136 = r14120 * r14120;
        double r14137 = r14130 * r14124;
        double r14138 = r14136 - r14137;
        double r14139 = sqrt(r14138);
        double r14140 = r14135 + r14139;
        double r14141 = r14134 / r14140;
        double r14142 = 6.18485615157459e+127;
        bool r14143 = r14120 <= r14142;
        double r14144 = r14135 - r14139;
        double r14145 = r14144 * r14131;
        double r14146 = 0.5;
        double r14147 = r14146 * r14125;
        double r14148 = 2.0;
        double r14149 = r14120 / r14130;
        double r14150 = r14148 * r14149;
        double r14151 = r14147 - r14150;
        double r14152 = r14143 ? r14145 : r14151;
        double r14153 = r14128 ? r14141 : r14152;
        double r14154 = r14122 ? r14126 : r14153;
        return r14154;
}

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 < -2.3088264856711725e+71

    1. Initial program 58.2

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

    2. Taylor expanded around -inf 3.1

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

    if -2.3088264856711725e+71 < b_2 < -7.104724366101806e-302

    1. Initial program 31.8

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

    3. Applied div-inv31.9

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

    5. Applied flip--31.9

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

    6. Applied associate-*l/31.9

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

    7. Simplified17.1

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

    8. Taylor expanded around 0 9.7

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

    if -7.104724366101806e-302 < b_2 < 6.18485615157459e+127

    1. Initial program 9.0

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

    3. Applied div-inv9.1

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

    if 6.18485615157459e+127 < b_2

    1. Initial program 55.4

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

    2. Taylor expanded around inf 3.1

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.30882648567117255 \cdot 10^{71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.1047243661018065 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.1848561515745898 \cdot 10^{127}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))