Average Error: 33.3 → 9.9
Time: 21.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot \frac{3}{2}\right)\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\
\;\;\;\;\frac{1}{3 \cdot a} \cdot \mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot \frac{3}{2}\right)\\

\mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\
\;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r3421464 = b;
        double r3421465 = -r3421464;
        double r3421466 = r3421464 * r3421464;
        double r3421467 = 3.0;
        double r3421468 = a;
        double r3421469 = r3421467 * r3421468;
        double r3421470 = c;
        double r3421471 = r3421469 * r3421470;
        double r3421472 = r3421466 - r3421471;
        double r3421473 = sqrt(r3421472);
        double r3421474 = r3421465 + r3421473;
        double r3421475 = r3421474 / r3421469;
        return r3421475;
}
↓
double f(double a, double b, double c) {
        double r3421476 = b;
        double r3421477 = -4.170773079316174e+99;
        bool r3421478 = r3421476 <= r3421477;
        double r3421479 = 1.0;
        double r3421480 = 3.0;
        double r3421481 = a;
        double r3421482 = r3421480 * r3421481;
        double r3421483 = r3421479 / r3421482;
        double r3421484 = -2.0;
        double r3421485 = c;
        double r3421486 = r3421476 / r3421481;
        double r3421487 = r3421485 / r3421486;
        double r3421488 = 1.5;
        double r3421489 = r3421487 * r3421488;
        double r3421490 = fma(r3421476, r3421484, r3421489);
        double r3421491 = r3421483 * r3421490;
        double r3421492 = 3.0168583404714427e-66;
        bool r3421493 = r3421476 <= r3421492;
        double r3421494 = -r3421476;
        double r3421495 = r3421476 * r3421476;
        double r3421496 = r3421485 * r3421482;
        double r3421497 = r3421495 - r3421496;
        double r3421498 = sqrt(r3421497);
        double r3421499 = r3421494 + r3421498;
        double r3421500 = r3421483 * r3421499;
        double r3421501 = -0.5;
        double r3421502 = r3421485 / r3421476;
        double r3421503 = r3421501 * r3421502;
        double r3421504 = r3421493 ? r3421500 : r3421503;
        double r3421505 = r3421478 ? r3421491 : r3421504;
        return r3421505;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.170773079316174e+99

    1. Initial program 44.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv44.3

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    4. Taylor expanded around -inf 10.0

      \[\leadsto \color{blue}{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)} \cdot \frac{1}{3 \cdot a}\]

    5. Simplified3.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot \frac{3}{2}\right)} \cdot \frac{1}{3 \cdot a}\]

    if -4.170773079316174e+99 < b < 3.0168583404714427e-66

    1. Initial program 12.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv13.0

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    if 3.0168583404714427e-66 < b

    1. Initial program 53.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 8.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.170773079316174 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \mathsf{fma}\left(b, -2, \frac{c}{\frac{b}{a}} \cdot \frac{3}{2}\right)\\ \mathbf{elif}\;b \le 3.0168583404714427 \cdot 10^{-66}:\\ \;\;\;\;\frac{1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))