Average Error: 34.3 → 9.8
Time: 18.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\ \;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\
\;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1920982614230223.5:\\
\;\;\;\;\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 r23017 = b_2;
        double r23018 = -r23017;
        double r23019 = r23017 * r23017;
        double r23020 = a;
        double r23021 = c;
        double r23022 = r23020 * r23021;
        double r23023 = r23019 - r23022;
        double r23024 = sqrt(r23023);
        double r23025 = r23018 - r23024;
        double r23026 = r23025 / r23020;
        return r23026;
}
↓
double f(double a, double b_2, double c) {
        double r23027 = b_2;
        double r23028 = -7.359940312872037e+54;
        bool r23029 = r23027 <= r23028;
        double r23030 = -0.5;
        double r23031 = c;
        double r23032 = r23031 / r23027;
        double r23033 = r23030 * r23032;
        double r23034 = -1.3943257168792356e-154;
        bool r23035 = r23027 <= r23034;
        double r23036 = a;
        double r23037 = r23036 * r23031;
        double r23038 = r23027 * r23027;
        double r23039 = r23038 - r23037;
        double r23040 = sqrt(r23039);
        double r23041 = r23040 - r23027;
        double r23042 = r23037 / r23041;
        double r23043 = 1.0;
        double r23044 = r23043 / r23036;
        double r23045 = r23042 * r23044;
        double r23046 = 1920982614230223.5;
        bool r23047 = r23027 <= r23046;
        double r23048 = -r23027;
        double r23049 = r23048 - r23040;
        double r23050 = r23049 / r23036;
        double r23051 = 0.5;
        double r23052 = r23051 * r23032;
        double r23053 = 2.0;
        double r23054 = r23027 / r23036;
        double r23055 = r23053 * r23054;
        double r23056 = r23052 - r23055;
        double r23057 = r23047 ? r23050 : r23056;
        double r23058 = r23035 ? r23045 : r23057;
        double r23059 = r23029 ? r23033 : r23058;
        return r23059;
}

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 < -7.359940312872037e+54

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 3.8

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

    if -7.359940312872037e+54 < b_2 < -1.3943257168792356e-154

    1. Initial program 39.5

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

    3. Applied div-inv39.6

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

    5. Applied flip--39.6

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

    6. Simplified18.5

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

    7. Simplified18.5

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

    if -1.3943257168792356e-154 < b_2 < 1920982614230223.5

    1. Initial program 12.3

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

    3. Applied div-inv12.4

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

    5. Applied un-div-inv12.3

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

    if 1920982614230223.5 < b_2

    1. Initial program 34.0

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

    2. Taylor expanded around inf 6.8

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.394325716879235550939300459885270666443 \cdot 10^{-154}:\\ \;\;\;\;\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1920982614230223.5:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))