Average Error: 34.5 → 10.7
Time: 19.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{3}}{a}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(\left(c \cdot a\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) \cdot \sqrt[3]{3}} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{3}}{a}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(\left(c \cdot a\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) \cdot \sqrt[3]{3}} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4699191 = b;
        double r4699192 = -r4699191;
        double r4699193 = r4699191 * r4699191;
        double r4699194 = 3.0;
        double r4699195 = a;
        double r4699196 = r4699194 * r4699195;
        double r4699197 = c;
        double r4699198 = r4699196 * r4699197;
        double r4699199 = r4699193 - r4699198;
        double r4699200 = sqrt(r4699199);
        double r4699201 = r4699192 + r4699200;
        double r4699202 = r4699201 / r4699196;
        return r4699202;
}
↓
double f(double a, double b, double c) {
        double r4699203 = b;
        double r4699204 = -2.221067196710922e+149;
        bool r4699205 = r4699203 <= r4699204;
        double r4699206 = 1.5;
        double r4699207 = c;
        double r4699208 = a;
        double r4699209 = r4699203 / r4699208;
        double r4699210 = r4699207 / r4699209;
        double r4699211 = -2.0;
        double r4699212 = r4699203 * r4699211;
        double r4699213 = fma(r4699206, r4699210, r4699212);
        double r4699214 = 3.0;
        double r4699215 = r4699213 / r4699214;
        double r4699216 = r4699215 / r4699208;
        double r4699217 = 2.8983489306952693e-35;
        bool r4699218 = r4699203 <= r4699217;
        double r4699219 = r4699203 * r4699203;
        double r4699220 = r4699207 * r4699208;
        double r4699221 = cbrt(r4699214);
        double r4699222 = r4699221 * r4699221;
        double r4699223 = r4699220 * r4699222;
        double r4699224 = r4699223 * r4699221;
        double r4699225 = r4699219 - r4699224;
        double r4699226 = sqrt(r4699225);
        double r4699227 = r4699226 - r4699203;
        double r4699228 = r4699227 / r4699214;
        double r4699229 = r4699228 / r4699208;
        double r4699230 = -0.5;
        double r4699231 = r4699207 / r4699203;
        double r4699232 = r4699230 * r4699231;
        double r4699233 = r4699218 ? r4699229 : r4699232;
        double r4699234 = r4699205 ? r4699216 : r4699233;
        return r4699234;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    3. Applied associate-/r*62.3

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

    4. Simplified62.3

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

    5. Taylor expanded around -inf 11.1

      \[\leadsto \frac{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified3.0

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, -2 \cdot b\right)}}{3}}{a}\]

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.7

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

    3. Applied associate-/r*14.7

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

    4. Simplified14.7

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

    6. Applied add-cube-cbrt14.7

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

    7. Applied associate-*r*14.7

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Taylor expanded around inf 7.3

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{3}}{a}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(\left(c \cdot a\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)\right) \cdot \sqrt[3]{3}} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))