Average Error: 34.5 → 6.9
Time: 15.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7604193036648139441831936:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}}\\ \mathbf{elif}\;b_2 \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\ \;\;\;\;\frac{-c}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7604193036648139441831936:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}}\\

\mathbf{elif}\;b_2 \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\
\;\;\;\;\frac{-c}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} + b_2}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r29435 = b_2;
        double r29436 = -r29435;
        double r29437 = r29435 * r29435;
        double r29438 = a;
        double r29439 = c;
        double r29440 = r29438 * r29439;
        double r29441 = r29437 - r29440;
        double r29442 = sqrt(r29441);
        double r29443 = r29436 + r29442;
        double r29444 = r29443 / r29438;
        return r29444;
}
↓
double f(double a, double b_2, double c) {
        double r29445 = b_2;
        double r29446 = -7.604193036648139e+24;
        bool r29447 = r29445 <= r29446;
        double r29448 = c;
        double r29449 = r29448 / r29445;
        double r29450 = 0.5;
        double r29451 = a;
        double r29452 = r29445 / r29451;
        double r29453 = -2.0;
        double r29454 = r29452 * r29453;
        double r29455 = fma(r29449, r29450, r29454);
        double r29456 = -2.1209008810311313e-243;
        bool r29457 = r29445 <= r29456;
        double r29458 = 1.0;
        double r29459 = -r29448;
        double r29460 = r29445 * r29445;
        double r29461 = fma(r29451, r29459, r29460);
        double r29462 = sqrt(r29461);
        double r29463 = r29462 - r29445;
        double r29464 = r29451 / r29463;
        double r29465 = r29458 / r29464;
        double r29466 = 2.3453700250865973e+84;
        bool r29467 = r29445 <= r29466;
        double r29468 = r29462 + r29445;
        double r29469 = r29459 / r29468;
        double r29470 = -0.5;
        double r29471 = r29470 * r29448;
        double r29472 = r29471 / r29445;
        double r29473 = r29467 ? r29469 : r29472;
        double r29474 = r29457 ? r29465 : r29473;
        double r29475 = r29447 ? r29455 : r29474;
        return r29475;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -7.604193036648139e+24

    1. Initial program 35.7

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]

    if -7.604193036648139e+24 < b_2 < -2.1209008810311313e-243

    1. Initial program 9.3

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

    3. Applied clear-num9.5

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

    4. Simplified9.5

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

    if -2.1209008810311313e-243 < b_2 < 2.3453700250865973e+84

    1. Initial program 29.5

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

    3. Applied flip-+29.6

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

    4. Simplified15.9

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

    5. Simplified15.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{-\left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}\right)}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity15.9

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

    8. Applied neg-mul-115.9

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

    9. Applied *-un-lft-identity15.9

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

    10. Applied times-frac15.9

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

    11. Applied times-frac15.9

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

    12. Simplified15.9

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

    13. Simplified15.1

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{c \cdot a}{a}}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}}}\]
    14. Using strategy rm

    15. Applied *-un-lft-identity15.1

      \[\leadsto -1 \cdot \frac{\frac{c \cdot a}{a}}{\color{blue}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}\right)}}\]

    16. Applied *-un-lft-identity15.1

      \[\leadsto -1 \cdot \frac{\color{blue}{1 \cdot \frac{c \cdot a}{a}}}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}\right)}\]

    17. Applied times-frac15.1

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{\frac{c \cdot a}{a}}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}}\right)}\]

    18. Simplified15.1

      \[\leadsto -1 \cdot \left(\color{blue}{1} \cdot \frac{\frac{c \cdot a}{a}}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, {b_2}^{2}\right)}}\right)\]

    19. Simplified9.3

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

    if 2.3453700250865973e+84 < b_2

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.5

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

    3. Simplified2.5

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7604193036648139441831936:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -2.120900881031131292062715264701944285734 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} - b_2}}\\ \mathbf{elif}\;b_2 \le 2.345370025086597272923559832061889684617 \cdot 10^{84}:\\ \;\;\;\;\frac{-c}{\sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))