Average Error: 43.8 → 11.1
Time: 7.9s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.004205915758382604285881001260349876247346:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.004205915758382604285881001260349876247346:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r114390 = b;
        double r114391 = -r114390;
        double r114392 = r114390 * r114390;
        double r114393 = 3.0;
        double r114394 = a;
        double r114395 = r114393 * r114394;
        double r114396 = c;
        double r114397 = r114395 * r114396;
        double r114398 = r114392 - r114397;
        double r114399 = sqrt(r114398);
        double r114400 = r114391 + r114399;
        double r114401 = r114400 / r114395;
        return r114401;
}
↓
double f(double a, double b, double c) {
        double r114402 = b;
        double r114403 = 0.004205915758382604;
        bool r114404 = r114402 <= r114403;
        double r114405 = r114402 * r114402;
        double r114406 = 3.0;
        double r114407 = a;
        double r114408 = r114406 * r114407;
        double r114409 = c;
        double r114410 = r114408 * r114409;
        double r114411 = fma(r114402, r114402, r114410);
        double r114412 = r114405 - r114411;
        double r114413 = r114405 - r114410;
        double r114414 = sqrt(r114413);
        double r114415 = r114402 + r114414;
        double r114416 = r114412 / r114415;
        double r114417 = r114416 / r114408;
        double r114418 = -0.5;
        double r114419 = r114409 / r114402;
        double r114420 = r114418 * r114419;
        double r114421 = r114404 ? r114417 : r114420;
        return r114421;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.004205915758382604

    1. Initial program 20.5

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

    2. Simplified20.5

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

    4. Applied flip--20.5

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

    5. Simplified19.7

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

    6. Simplified19.7

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

    if 0.004205915758382604 < b

    1. Initial program 46.3

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

    2. Simplified46.3

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

    3. Taylor expanded around inf 10.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.004205915758382604285881001260349876247346:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))