Average Error: 33.1 → 8.8
Time: 16.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.657184583194415 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot \frac{3}{2}}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 1.4458951003956485 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.6715468005080632 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
\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 -5.657184583194415 \cdot 10^{+105}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot \frac{3}{2}}{\frac{b}{c}}\right)}{3}}{a}\\

\mathbf{elif}\;b \le 1.4458951003956485 \cdot 10^{-170}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + \left(-b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.6715468005080632 \cdot 10^{+36}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4727469 = b;
        double r4727470 = -r4727469;
        double r4727471 = r4727469 * r4727469;
        double r4727472 = 3.0;
        double r4727473 = a;
        double r4727474 = r4727472 * r4727473;
        double r4727475 = c;
        double r4727476 = r4727474 * r4727475;
        double r4727477 = r4727471 - r4727476;
        double r4727478 = sqrt(r4727477);
        double r4727479 = r4727470 + r4727478;
        double r4727480 = r4727479 / r4727474;
        return r4727480;
}
↓
double f(double a, double b, double c) {
        double r4727481 = b;
        double r4727482 = -5.657184583194415e+105;
        bool r4727483 = r4727481 <= r4727482;
        double r4727484 = -2.0;
        double r4727485 = a;
        double r4727486 = 1.5;
        double r4727487 = r4727485 * r4727486;
        double r4727488 = c;
        double r4727489 = r4727481 / r4727488;
        double r4727490 = r4727487 / r4727489;
        double r4727491 = fma(r4727484, r4727481, r4727490);
        double r4727492 = 3.0;
        double r4727493 = r4727491 / r4727492;
        double r4727494 = r4727493 / r4727485;
        double r4727495 = 1.4458951003956485e-170;
        bool r4727496 = r4727481 <= r4727495;
        double r4727497 = -3.0;
        double r4727498 = r4727485 * r4727497;
        double r4727499 = r4727488 * r4727498;
        double r4727500 = fma(r4727481, r4727481, r4727499);
        double r4727501 = sqrt(r4727500);
        double r4727502 = -r4727481;
        double r4727503 = r4727501 + r4727502;
        double r4727504 = r4727492 * r4727485;
        double r4727505 = r4727503 / r4727504;
        double r4727506 = 1.6715468005080632e+36;
        bool r4727507 = r4727481 <= r4727506;
        double r4727508 = 0.0;
        double r4727509 = fma(r4727488, r4727498, r4727508);
        double r4727510 = r4727481 + r4727501;
        double r4727511 = r4727509 / r4727510;
        double r4727512 = r4727511 / r4727492;
        double r4727513 = r4727512 / r4727485;
        double r4727514 = r4727488 / r4727481;
        double r4727515 = -0.5;
        double r4727516 = r4727514 * r4727515;
        double r4727517 = r4727507 ? r4727513 : r4727516;
        double r4727518 = r4727496 ? r4727505 : r4727517;
        double r4727519 = r4727483 ? r4727494 : r4727518;
        return r4727519;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -5.657184583194415e+105

    1. Initial program 46.0

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

    3. Applied associate-/r*46.0

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

    4. Simplified46.0

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

    5. Taylor expanded around -inf 10.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified4.4

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

    if -5.657184583194415e+105 < b < 1.4458951003956485e-170

    1. Initial program 10.6

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

    2. Taylor expanded around 0 10.6

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

    3. Simplified10.6

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

    if 1.4458951003956485e-170 < b < 1.6715468005080632e+36

    1. Initial program 34.2

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

    3. Applied associate-/r*34.3

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

    4. Simplified34.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} - b}{3}}}{a}\]
    5. Using strategy rm

    6. Applied flip--34.4

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

    7. Simplified16.7

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(c, -3 \cdot a, 0\right)}}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-3 \cdot a\right)\right)} + b}}{3}}{a}\]

    if 1.6715468005080632e+36 < b

    1. Initial program 56.4

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

    2. Taylor expanded around inf 4.2

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.657184583194415 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot \frac{3}{2}}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 1.4458951003956485 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} + \left(-b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.6715468005080632 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))