Average Error: 33.9 → 6.7
Time: 22.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r29490 = b_2;
        double r29491 = -r29490;
        double r29492 = r29490 * r29490;
        double r29493 = a;
        double r29494 = c;
        double r29495 = r29493 * r29494;
        double r29496 = r29492 - r29495;
        double r29497 = sqrt(r29496);
        double r29498 = r29491 - r29497;
        double r29499 = r29498 / r29493;
        return r29499;
}
↓
double f(double a, double b_2, double c) {
        double r29500 = b_2;
        double r29501 = -5.674469085146397e+110;
        bool r29502 = r29500 <= r29501;
        double r29503 = -0.5;
        double r29504 = c;
        double r29505 = r29504 / r29500;
        double r29506 = r29503 * r29505;
        double r29507 = 1.4099136946117732e-265;
        bool r29508 = r29500 <= r29507;
        double r29509 = r29500 * r29500;
        double r29510 = a;
        double r29511 = r29510 * r29504;
        double r29512 = r29509 - r29511;
        double r29513 = sqrt(r29512);
        double r29514 = r29513 - r29500;
        double r29515 = r29504 / r29514;
        double r29516 = 1.7151811081882383e+78;
        bool r29517 = r29500 <= r29516;
        double r29518 = -r29500;
        double r29519 = r29518 - r29513;
        double r29520 = 1.0;
        double r29521 = r29520 / r29510;
        double r29522 = r29519 * r29521;
        double r29523 = -2.0;
        double r29524 = r29523 * r29500;
        double r29525 = r29524 / r29510;
        double r29526 = r29517 ? r29522 : r29525;
        double r29527 = r29508 ? r29515 : r29526;
        double r29528 = r29502 ? r29506 : r29527;
        return r29528;
}

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 < -5.674469085146397e+110

    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 2.7

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

    if -5.674469085146397e+110 < b_2 < 1.4099136946117732e-265

    1. Initial program 31.8

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

    3. Applied flip--31.8

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied *-un-lft-identity16.1

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

    9. Applied times-frac16.1

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

    10. Simplified16.1

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

    11. Simplified8.7

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

    if 1.4099136946117732e-265 < b_2 < 1.7151811081882383e+78

    1. Initial program 8.5

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

    3. Applied div-inv8.6

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

    if 1.7151811081882383e+78 < b_2

    1. Initial program 43.0

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

    3. Applied flip--62.6

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    6. Taylor expanded around 0 4.8

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.409913694611773174476543925182798865564 \cdot 10^{-265}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))