Average Error: 33.7 → 11.6
Time: 22.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8811095525294671235916146659402186752:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 0.04537546925850252654832672760676359757781:\\ \;\;\;\;\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \frac{1}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8811095525294671235916146659402186752:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 0.04537546925850252654832672760676359757781:\\
\;\;\;\;\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \frac{1}{a} - \frac{b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r2095395 = b_2;
        double r2095396 = -r2095395;
        double r2095397 = r2095395 * r2095395;
        double r2095398 = a;
        double r2095399 = c;
        double r2095400 = r2095398 * r2095399;
        double r2095401 = r2095397 - r2095400;
        double r2095402 = sqrt(r2095401);
        double r2095403 = r2095396 + r2095402;
        double r2095404 = r2095403 / r2095398;
        return r2095404;
}
↓
double f(double a, double b_2, double c) {
        double r2095405 = b_2;
        double r2095406 = -8.811095525294671e+36;
        bool r2095407 = r2095405 <= r2095406;
        double r2095408 = 0.5;
        double r2095409 = c;
        double r2095410 = r2095409 / r2095405;
        double r2095411 = r2095408 * r2095410;
        double r2095412 = 2.0;
        double r2095413 = a;
        double r2095414 = r2095405 / r2095413;
        double r2095415 = r2095412 * r2095414;
        double r2095416 = r2095411 - r2095415;
        double r2095417 = 0.04537546925850253;
        bool r2095418 = r2095405 <= r2095417;
        double r2095419 = r2095405 * r2095405;
        double r2095420 = r2095413 * r2095409;
        double r2095421 = r2095419 - r2095420;
        double r2095422 = sqrt(r2095421);
        double r2095423 = 1.0;
        double r2095424 = r2095423 / r2095413;
        double r2095425 = r2095422 * r2095424;
        double r2095426 = r2095425 - r2095414;
        double r2095427 = -0.5;
        double r2095428 = r2095427 * r2095410;
        double r2095429 = r2095418 ? r2095426 : r2095428;
        double r2095430 = r2095407 ? r2095416 : r2095429;
        return r2095430;
}

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 3 regimes

  2. if b_2 < -8.811095525294671e+36

    1. Initial program 36.4

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

    2. Simplified36.4

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

    3. Taylor expanded around -inf 6.1

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

    if -8.811095525294671e+36 < b_2 < 0.04537546925850253

    1. Initial program 17.5

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

    2. Simplified17.5

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

    4. Applied div-sub17.5

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

    6. Applied div-inv17.6

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

    if 0.04537546925850253 < b_2

    1. Initial program 54.9

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

    2. Simplified54.9

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

    3. Taylor expanded around inf 6.4

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8811095525294671235916146659402186752:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 0.04537546925850252654832672760676359757781:\\ \;\;\;\;\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \frac{1}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))