Average Error: 34.4 → 7.7
Time: 40.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.661523975777196 \cdot 10^{-253}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.7205132563858266 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.661523975777196 \cdot 10^{-253}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1.7205132563858266 \cdot 10^{+103}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r995478 = b_2;
        double r995479 = -r995478;
        double r995480 = r995478 * r995478;
        double r995481 = a;
        double r995482 = c;
        double r995483 = r995481 * r995482;
        double r995484 = r995480 - r995483;
        double r995485 = sqrt(r995484);
        double r995486 = r995479 - r995485;
        double r995487 = r995486 / r995481;
        return r995487;
}
↓
double f(double a, double b_2, double c) {
        double r995488 = b_2;
        double r995489 = -3.234164035284793e+22;
        bool r995490 = r995488 <= r995489;
        double r995491 = -0.5;
        double r995492 = c;
        double r995493 = r995492 / r995488;
        double r995494 = r995491 * r995493;
        double r995495 = 2.661523975777196e-253;
        bool r995496 = r995488 <= r995495;
        double r995497 = a;
        double r995498 = r995488 * r995488;
        double r995499 = r995492 * r995497;
        double r995500 = r995498 - r995499;
        double r995501 = sqrt(r995500);
        double r995502 = r995501 - r995488;
        double r995503 = r995492 / r995502;
        double r995504 = r995497 * r995503;
        double r995505 = r995504 / r995497;
        double r995506 = 1.7205132563858266e+103;
        bool r995507 = r995488 <= r995506;
        double r995508 = 1.0;
        double r995509 = r995508 / r995497;
        double r995510 = -r995488;
        double r995511 = r995510 - r995501;
        double r995512 = r995509 * r995511;
        double r995513 = 0.5;
        double r995514 = r995493 * r995513;
        double r995515 = 2.0;
        double r995516 = r995488 / r995497;
        double r995517 = r995515 * r995516;
        double r995518 = r995514 - r995517;
        double r995519 = r995507 ? r995512 : r995518;
        double r995520 = r995496 ? r995505 : r995519;
        double r995521 = r995490 ? r995494 : r995520;
        return r995521;
}

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 < -3.234164035284793e+22

    1. Initial program 56.6

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

    2. Taylor expanded around -inf 4.6

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

    if -3.234164035284793e+22 < b_2 < 2.661523975777196e-253

    1. Initial program 26.0

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

    3. Applied flip--26.1

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

    4. Simplified16.3

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

    5. Simplified16.3

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

    7. Applied *-un-lft-identity16.3

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

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

    if 2.661523975777196e-253 < b_2 < 1.7205132563858266e+103

    1. Initial program 8.3

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

    3. Applied div-inv8.4

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

    if 1.7205132563858266e+103 < b_2

    1. Initial program 47.4

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.661523975777196 \cdot 10^{-253}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.7205132563858266 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))