Average Error: 33.9 → 6.8
Time: 18.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\
\;\;\;\;\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 r22474 = b_2;
        double r22475 = -r22474;
        double r22476 = r22474 * r22474;
        double r22477 = a;
        double r22478 = c;
        double r22479 = r22477 * r22478;
        double r22480 = r22476 - r22479;
        double r22481 = sqrt(r22480);
        double r22482 = r22475 - r22481;
        double r22483 = r22482 / r22477;
        return r22483;
}
↓
double f(double a, double b_2, double c) {
        double r22484 = b_2;
        double r22485 = -3.359953003549157e+103;
        bool r22486 = r22484 <= r22485;
        double r22487 = -0.5;
        double r22488 = c;
        double r22489 = r22488 / r22484;
        double r22490 = r22487 * r22489;
        double r22491 = 2.094358742794728e-239;
        bool r22492 = r22484 <= r22491;
        double r22493 = -r22484;
        double r22494 = r22484 * r22484;
        double r22495 = a;
        double r22496 = r22495 * r22488;
        double r22497 = r22494 - r22496;
        double r22498 = sqrt(r22497);
        double r22499 = r22493 + r22498;
        double r22500 = r22488 / r22499;
        double r22501 = 5.099089738165329e+67;
        bool r22502 = r22484 <= r22501;
        double r22503 = r22493 - r22498;
        double r22504 = r22503 / r22495;
        double r22505 = 0.5;
        double r22506 = r22505 * r22489;
        double r22507 = 2.0;
        double r22508 = r22484 / r22495;
        double r22509 = r22507 * r22508;
        double r22510 = r22506 - r22509;
        double r22511 = r22502 ? r22504 : r22510;
        double r22512 = r22492 ? r22500 : r22511;
        double r22513 = r22486 ? r22490 : r22512;
        return r22513;
}

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 < -3.359953003549157e+103

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.5

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

    if -3.359953003549157e+103 < b_2 < 2.094358742794728e-239

    1. Initial program 30.6

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

    3. Applied clear-num30.6

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

    5. Applied flip--30.7

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

    6. Applied associate-/r/30.8

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

    7. Applied associate-/r*30.8

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

    8. Simplified15.4

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

    9. Taylor expanded around 0 9.5

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

    if 2.094358742794728e-239 < b_2 < 5.099089738165329e+67

    1. Initial program 8.0

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

    if 5.099089738165329e+67 < b_2

    1. Initial program 40.4

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

    2. Taylor expanded around inf 5.4

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))