Average Error: 33.7 → 13.1
Time: 8.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.26967329570505124 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(\sqrt[3]{c \cdot \left(3 \cdot a\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{c \cdot \left(3 \cdot a\right)}\right)}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -2.2724541866372811 \cdot 10^{165}:\\
\;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\

\mathbf{elif}\;b \le -5.26967329570505124 \cdot 10^{-298}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(\sqrt[3]{c \cdot \left(3 \cdot a\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{c \cdot \left(3 \cdot a\right)}\right)}}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r163217 = b;
        double r163218 = -r163217;
        double r163219 = r163217 * r163217;
        double r163220 = 3.0;
        double r163221 = a;
        double r163222 = r163220 * r163221;
        double r163223 = c;
        double r163224 = r163222 * r163223;
        double r163225 = r163219 - r163224;
        double r163226 = sqrt(r163225);
        double r163227 = r163218 + r163226;
        double r163228 = r163227 / r163222;
        return r163228;
}
↓
double f(double a, double b, double c) {
        double r163229 = b;
        double r163230 = -2.272454186637281e+165;
        bool r163231 = r163229 <= r163230;
        double r163232 = -r163229;
        double r163233 = 1.5;
        double r163234 = a;
        double r163235 = c;
        double r163236 = r163234 * r163235;
        double r163237 = r163236 / r163229;
        double r163238 = r163233 * r163237;
        double r163239 = r163238 - r163229;
        double r163240 = r163232 + r163239;
        double r163241 = 3.0;
        double r163242 = r163241 * r163234;
        double r163243 = r163240 / r163242;
        double r163244 = -5.269673295705051e-298;
        bool r163245 = r163229 <= r163244;
        double r163246 = r163235 * r163242;
        double r163247 = cbrt(r163246);
        double r163248 = cbrt(r163241);
        double r163249 = cbrt(r163236);
        double r163250 = r163248 * r163249;
        double r163251 = r163247 * r163250;
        double r163252 = r163251 * r163247;
        double r163253 = -r163252;
        double r163254 = fma(r163229, r163229, r163253);
        double r163255 = sqrt(r163254);
        double r163256 = r163232 + r163255;
        double r163257 = r163256 / r163242;
        double r163258 = 1.0331115085790278e+154;
        bool r163259 = r163229 <= r163258;
        double r163260 = -r163246;
        double r163261 = fma(r163229, r163229, r163260);
        double r163262 = sqrt(r163261);
        double r163263 = r163232 - r163262;
        double r163264 = r163246 / r163263;
        double r163265 = r163264 / r163242;
        double r163266 = -1.5;
        double r163267 = r163266 * r163237;
        double r163268 = r163267 / r163242;
        double r163269 = r163259 ? r163265 : r163268;
        double r163270 = r163245 ? r163257 : r163269;
        double r163271 = r163231 ? r163243 : r163270;
        return r163271;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -2.272454186637281e+165

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 10.5

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

    if -2.272454186637281e+165 < b < -5.269673295705051e-298

    1. Initial program 9.5

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

    2. Taylor expanded around 0 9.5

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

    3. Simplified9.5

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

    5. Applied add-cube-cbrt9.7

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

    6. Taylor expanded around 0 56.3

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(\sqrt[3]{c \cdot \left(3 \cdot a\right)} \cdot \color{blue}{\left(\sqrt[3]{3} \cdot e^{\frac{1}{3} \cdot \left(\log a + \log c\right)}\right)}\right) \cdot \sqrt[3]{c \cdot \left(3 \cdot a\right)}\right)}}{3 \cdot a}\]

    7. Simplified9.7

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

    if -5.269673295705051e-298 < b < 1.0331115085790278e+154

    1. Initial program 33.9

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

    2. Taylor expanded around 0 34.0

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

    3. Simplified33.9

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

    5. Applied flip-+33.9

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

    6. Simplified16.3

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

    if 1.0331115085790278e+154 < b

    1. Initial program 64.0

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

    2. Taylor expanded around inf 14.5

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.26967329570505124 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(\sqrt[3]{c \cdot \left(3 \cdot a\right)} \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{a \cdot c}\right)\right) \cdot \sqrt[3]{c \cdot \left(3 \cdot a\right)}\right)}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{c \cdot \left(3 \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))