Average Error: 44.1 → 11.3
Time: 12.5s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.1755425336568303684714464907301589846611:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.1755425336568303684714464907301589846611:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r81365 = b;
        double r81366 = -r81365;
        double r81367 = r81365 * r81365;
        double r81368 = 3.0;
        double r81369 = a;
        double r81370 = r81368 * r81369;
        double r81371 = c;
        double r81372 = r81370 * r81371;
        double r81373 = r81367 - r81372;
        double r81374 = sqrt(r81373);
        double r81375 = r81366 + r81374;
        double r81376 = r81375 / r81370;
        return r81376;
}
↓
double f(double a, double b, double c) {
        double r81377 = b;
        double r81378 = 0.17554253365683037;
        bool r81379 = r81377 <= r81378;
        double r81380 = 3.0;
        double r81381 = a;
        double r81382 = r81380 * r81381;
        double r81383 = c;
        double r81384 = r81382 * r81383;
        double r81385 = -r81384;
        double r81386 = fma(r81377, r81377, r81385);
        double r81387 = sqrt(r81386);
        double r81388 = r81387 - r81377;
        double r81389 = r81388 / r81382;
        double r81390 = -1.5;
        double r81391 = r81390 / r81380;
        double r81392 = r81383 / r81377;
        double r81393 = r81391 * r81392;
        double r81394 = r81379 ? r81389 : r81393;
        return r81394;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.17554253365683037

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied fma-neg23.2

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

    if 0.17554253365683037 < b

    1. Initial program 47.5

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

    2. Simplified47.5

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

    3. Taylor expanded around inf 9.7

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied times-frac9.6

      \[\leadsto \color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a \cdot c}{b}}{a}}\]

    6. Taylor expanded around 0 9.4

      \[\leadsto \frac{-1.5}{3} \cdot \color{blue}{\frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.1755425336568303684714464907301589846611:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5}{3} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))