Average Error: 34.0 → 6.8
Time: 19.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\ \;\;\;\;\frac{c}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\ \mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\
\;\;\;\;\frac{c}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\

\mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r28367 = b_2;
        double r28368 = -r28367;
        double r28369 = r28367 * r28367;
        double r28370 = a;
        double r28371 = c;
        double r28372 = r28370 * r28371;
        double r28373 = r28369 - r28372;
        double r28374 = sqrt(r28373);
        double r28375 = r28368 - r28374;
        double r28376 = r28375 / r28370;
        return r28376;
}
↓
double f(double a, double b_2, double c) {
        double r28377 = b_2;
        double r28378 = -3.7277239731487426e+105;
        bool r28379 = r28377 <= r28378;
        double r28380 = -0.5;
        double r28381 = c;
        double r28382 = r28381 / r28377;
        double r28383 = r28380 * r28382;
        double r28384 = 1.3645292833571162e-246;
        bool r28385 = r28377 <= r28384;
        double r28386 = a;
        double r28387 = r28381 * r28386;
        double r28388 = -r28387;
        double r28389 = fma(r28377, r28377, r28388);
        double r28390 = sqrt(r28389);
        double r28391 = r28390 - r28377;
        double r28392 = r28381 / r28391;
        double r28393 = 4.19865099347443e+74;
        bool r28394 = r28377 <= r28393;
        double r28395 = -r28377;
        double r28396 = r28395 / r28386;
        double r28397 = r28377 * r28377;
        double r28398 = r28386 * r28381;
        double r28399 = r28397 - r28398;
        double r28400 = sqrt(r28399);
        double r28401 = r28400 / r28386;
        double r28402 = r28396 - r28401;
        double r28403 = 0.5;
        double r28404 = r28377 / r28386;
        double r28405 = -2.0;
        double r28406 = r28404 * r28405;
        double r28407 = fma(r28382, r28403, r28406);
        double r28408 = r28394 ? r28402 : r28407;
        double r28409 = r28385 ? r28392 : r28408;
        double r28410 = r28379 ? r28383 : r28409;
        return r28410;
}

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 < -3.7277239731487426e+105

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 2.4

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

    if -3.7277239731487426e+105 < b_2 < 1.3645292833571162e-246

    1. Initial program 30.7

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

    3. Applied flip--30.8

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

    4. Simplified15.7

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

    5. Simplified15.7

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

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

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

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

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

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

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

    10. Applied times-frac15.7

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

    11. Applied times-frac15.7

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

    12. Simplified15.7

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

    13. Simplified8.5

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

    if 1.3645292833571162e-246 < b_2 < 4.19865099347443e+74

    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 div-sub9.3

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

    if 4.19865099347443e+74 < b_2

    1. Initial program 41.8

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

    2. Taylor expanded around inf 5.3

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

    3. Simplified5.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.727723973148742563324153150500151191249 \cdot 10^{105}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.364529283357116165462040672443911208369 \cdot 10^{-246}:\\ \;\;\;\;\frac{c}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\ \mathbf{elif}\;b_2 \le 4.198650993474429809884798868723112488797 \cdot 10^{74}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))