Average Error: 34.2 → 8.9
Time: 18.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\
\;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\

\mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\
\;\;\;\;\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 r67984 = b_2;
        double r67985 = -r67984;
        double r67986 = r67984 * r67984;
        double r67987 = a;
        double r67988 = c;
        double r67989 = r67987 * r67988;
        double r67990 = r67986 - r67989;
        double r67991 = sqrt(r67990);
        double r67992 = r67985 - r67991;
        double r67993 = r67992 / r67987;
        return r67993;
}
↓
double f(double a, double b_2, double c) {
        double r67994 = b_2;
        double r67995 = -1.2900385447307874e+100;
        bool r67996 = r67994 <= r67995;
        double r67997 = -0.5;
        double r67998 = c;
        double r67999 = r67998 / r67994;
        double r68000 = r67997 * r67999;
        double r68001 = 1.0711388394770918e-291;
        bool r68002 = r67994 <= r68001;
        double r68003 = a;
        double r68004 = r67994 * r67994;
        double r68005 = r68003 * r67998;
        double r68006 = r68004 - r68005;
        double r68007 = sqrt(r68006);
        double r68008 = r68007 - r67994;
        double r68009 = sqrt(r68008);
        double r68010 = r68009 / r67998;
        double r68011 = r68003 / r68010;
        double r68012 = r68011 / r68009;
        double r68013 = r68012 / r68003;
        double r68014 = 1.4947314724287081e+91;
        bool r68015 = r67994 <= r68014;
        double r68016 = -r67994;
        double r68017 = r68016 - r68007;
        double r68018 = 1.0;
        double r68019 = r68018 / r68003;
        double r68020 = r68017 * r68019;
        double r68021 = 0.5;
        double r68022 = r68021 * r67999;
        double r68023 = 2.0;
        double r68024 = r67994 / r68003;
        double r68025 = r68023 * r68024;
        double r68026 = r68022 - r68025;
        double r68027 = r68015 ? r68020 : r68026;
        double r68028 = r68002 ? r68013 : r68027;
        double r68029 = r67996 ? r68000 : r68028;
        return r68029;
}

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 < -1.2900385447307874e+100

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 2.6

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

    if -1.2900385447307874e+100 < b_2 < 1.0711388394770918e-291

    1. Initial program 31.7

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

    3. Applied flip--31.8

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

    4. Simplified17.3

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

    5. Simplified17.3

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

    7. Applied add-sqr-sqrt17.5

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

    8. Applied associate-/r*17.5

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

    9. Simplified16.0

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

    if 1.0711388394770918e-291 < b_2 < 1.4947314724287081e+91

    1. Initial program 8.7

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

    3. Applied div-inv8.8

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

    if 1.4947314724287081e+91 < b_2

    1. Initial program 45.7

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))