Average Error: 34.9 → 6.6
Time: 19.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\
\;\;\;\;\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 r26033 = b_2;
        double r26034 = -r26033;
        double r26035 = r26033 * r26033;
        double r26036 = a;
        double r26037 = c;
        double r26038 = r26036 * r26037;
        double r26039 = r26035 - r26038;
        double r26040 = sqrt(r26039);
        double r26041 = r26034 - r26040;
        double r26042 = r26041 / r26036;
        return r26042;
}
↓
double f(double a, double b_2, double c) {
        double r26043 = b_2;
        double r26044 = -1.8427455378627112e+150;
        bool r26045 = r26043 <= r26044;
        double r26046 = -0.5;
        double r26047 = c;
        double r26048 = r26047 / r26043;
        double r26049 = r26046 * r26048;
        double r26050 = -2.5157876682752365e-293;
        bool r26051 = r26043 <= r26050;
        double r26052 = r26043 * r26043;
        double r26053 = a;
        double r26054 = r26053 * r26047;
        double r26055 = r26052 - r26054;
        double r26056 = sqrt(r26055);
        double r26057 = r26056 - r26043;
        double r26058 = r26047 / r26057;
        double r26059 = 4.340805540534955e+107;
        bool r26060 = r26043 <= r26059;
        double r26061 = 1.0;
        double r26062 = -r26043;
        double r26063 = r26062 - r26056;
        double r26064 = r26053 / r26063;
        double r26065 = r26061 / r26064;
        double r26066 = 0.5;
        double r26067 = r26066 * r26048;
        double r26068 = 2.0;
        double r26069 = r26043 / r26053;
        double r26070 = r26068 * r26069;
        double r26071 = r26067 - r26070;
        double r26072 = r26060 ? r26065 : r26071;
        double r26073 = r26051 ? r26058 : r26072;
        double r26074 = r26045 ? r26049 : r26073;
        return r26074;
}

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 < -1.8427455378627112e+150

    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 1.7

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

    if -1.8427455378627112e+150 < b_2 < -2.5157876682752365e-293

    1. Initial program 35.8

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

    3. Applied flip--35.8

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    7. Applied div-inv17.0

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

    9. Applied associate-*l/15.3

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

    10. Simplified15.2

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

    11. Taylor expanded around 0 8.2

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

    if -2.5157876682752365e-293 < b_2 < 4.340805540534955e+107

    1. Initial program 9.4

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

    3. Applied clear-num9.5

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

    if 4.340805540534955e+107 < b_2

    1. Initial program 49.7

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))