Average Error: 33.2 → 10.0
Time: 42.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.460834059807099 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 2.487495599193463 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.460834059807099 \cdot 10^{+151}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 2.487495599193463 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{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 r14051199 = b;
        double r14051200 = -r14051199;
        double r14051201 = r14051199 * r14051199;
        double r14051202 = 3.0;
        double r14051203 = a;
        double r14051204 = r14051202 * r14051203;
        double r14051205 = c;
        double r14051206 = r14051204 * r14051205;
        double r14051207 = r14051201 - r14051206;
        double r14051208 = sqrt(r14051207);
        double r14051209 = r14051200 + r14051208;
        double r14051210 = r14051209 / r14051204;
        return r14051210;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14051211 = b;
        double r14051212 = -2.460834059807099e+151;
        bool r14051213 = r14051211 <= r14051212;
        double r14051214 = 0.5;
        double r14051215 = c;
        double r14051216 = r14051215 / r14051211;
        double r14051217 = r14051214 * r14051216;
        double r14051218 = a;
        double r14051219 = r14051211 / r14051218;
        double r14051220 = 0.6666666666666666;
        double r14051221 = r14051219 * r14051220;
        double r14051222 = r14051217 - r14051221;
        double r14051223 = 2.487495599193463e-81;
        bool r14051224 = r14051211 <= r14051223;
        double r14051225 = r14051211 * r14051211;
        double r14051226 = 3.0;
        double r14051227 = r14051215 * r14051218;
        double r14051228 = r14051226 * r14051227;
        double r14051229 = r14051225 - r14051228;
        double r14051230 = sqrt(r14051229);
        double r14051231 = r14051230 - r14051211;
        double r14051232 = r14051231 / r14051226;
        double r14051233 = r14051232 / r14051218;
        double r14051234 = -0.5;
        double r14051235 = r14051234 * r14051216;
        double r14051236 = r14051224 ? r14051233 : r14051235;
        double r14051237 = r14051213 ? r14051222 : r14051236;
        return r14051237;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -2.460834059807099e+151

    1. Initial program 59.7

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

    2. Simplified59.7

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

    3. Taylor expanded around -inf 2.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    if -2.460834059807099e+151 < b < 2.487495599193463e-81

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied associate-/r*12.2

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

    5. Taylor expanded around inf 12.2

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

    if 2.487495599193463e-81 < b

    1. Initial program 52.1

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

    2. Simplified52.1

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.460834059807099 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 2.487495599193463 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))