Average Error: 32.8 → 10.5
Time: 34.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1148847116724585 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.1148847116724585 \cdot 10^{+104}:\\
\;\;\;\;\frac{\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3}}{a}\\

\mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r12844397 = b;
        double r12844398 = -r12844397;
        double r12844399 = r12844397 * r12844397;
        double r12844400 = 3.0;
        double r12844401 = a;
        double r12844402 = r12844400 * r12844401;
        double r12844403 = c;
        double r12844404 = r12844402 * r12844403;
        double r12844405 = r12844399 - r12844404;
        double r12844406 = sqrt(r12844405);
        double r12844407 = r12844398 + r12844406;
        double r12844408 = r12844407 / r12844402;
        return r12844408;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r12844409 = b;
        double r12844410 = -2.1148847116724585e+104;
        bool r12844411 = r12844409 <= r12844410;
        double r12844412 = 1.5;
        double r12844413 = a;
        double r12844414 = c;
        double r12844415 = r12844409 / r12844414;
        double r12844416 = r12844413 / r12844415;
        double r12844417 = -2.0;
        double r12844418 = r12844409 * r12844417;
        double r12844419 = fma(r12844412, r12844416, r12844418);
        double r12844420 = 3.0;
        double r12844421 = r12844419 / r12844420;
        double r12844422 = r12844421 / r12844413;
        double r12844423 = 1.0937455763637174e-150;
        bool r12844424 = r12844409 <= r12844423;
        double r12844425 = r12844409 * r12844409;
        double r12844426 = r12844420 * r12844413;
        double r12844427 = r12844414 * r12844426;
        double r12844428 = r12844425 - r12844427;
        double r12844429 = sqrt(r12844428);
        double r12844430 = -r12844409;
        double r12844431 = r12844429 + r12844430;
        double r12844432 = r12844431 / r12844420;
        double r12844433 = r12844432 / r12844413;
        double r12844434 = -0.5;
        double r12844435 = r12844414 / r12844409;
        double r12844436 = r12844434 * r12844435;
        double r12844437 = r12844424 ? r12844433 : r12844436;
        double r12844438 = r12844411 ? r12844422 : r12844437;
        return r12844438;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1148847116724585e+104

    1. Initial program 44.5

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

    3. Applied associate-/r*44.5

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

    4. Taylor expanded around -inf 9.9

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

    5. Simplified4.0

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

    if -2.1148847116724585e+104 < b < 1.0937455763637174e-150

    1. Initial program 10.3

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

    3. Applied associate-/r*10.3

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

    if 1.0937455763637174e-150 < b

    1. Initial program 49.4

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

    2. Taylor expanded around inf 12.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1148847116724585 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{(\frac{3}{2} \cdot \left(\frac{a}{\frac{b}{c}}\right) + \left(b \cdot -2\right))_*}{3}}{a}\\ \mathbf{elif}\;b \le 1.0937455763637174 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))