Average Error: 33.3 → 14.8
Time: 28.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.367565365770909 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.5222805660690795 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.367565365770909 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.5222805660690795 \cdot 10^{-59}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15515305 = b;
        double r15515306 = -r15515305;
        double r15515307 = r15515305 * r15515305;
        double r15515308 = 3.0;
        double r15515309 = a;
        double r15515310 = r15515308 * r15515309;
        double r15515311 = c;
        double r15515312 = r15515310 * r15515311;
        double r15515313 = r15515307 - r15515312;
        double r15515314 = sqrt(r15515313);
        double r15515315 = r15515306 + r15515314;
        double r15515316 = r15515315 / r15515310;
        return r15515316;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15515317 = b;
        double r15515318 = -1.367565365770909e+154;
        bool r15515319 = r15515317 <= r15515318;
        double r15515320 = 1.5;
        double r15515321 = a;
        double r15515322 = c;
        double r15515323 = r15515321 * r15515322;
        double r15515324 = r15515323 / r15515317;
        double r15515325 = r15515320 * r15515324;
        double r15515326 = r15515325 - r15515317;
        double r15515327 = r15515326 - r15515317;
        double r15515328 = 3.0;
        double r15515329 = r15515328 * r15515321;
        double r15515330 = r15515327 / r15515329;
        double r15515331 = 1.5222805660690795e-59;
        bool r15515332 = r15515317 <= r15515331;
        double r15515333 = r15515317 * r15515317;
        double r15515334 = r15515329 * r15515322;
        double r15515335 = r15515333 - r15515334;
        double r15515336 = sqrt(r15515335);
        double r15515337 = r15515336 - r15515317;
        double r15515338 = r15515337 / r15515329;
        double r15515339 = -1.5;
        double r15515340 = r15515339 * r15515324;
        double r15515341 = r15515340 / r15515329;
        double r15515342 = r15515332 ? r15515338 : r15515341;
        double r15515343 = r15515319 ? r15515330 : r15515342;
        return r15515343;
}

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 < -1.367565365770909e+154

    1. Initial program 61.0

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

    2. Simplified61.0

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

    3. Taylor expanded around -inf 11.6

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

    if -1.367565365770909e+154 < b < 1.5222805660690795e-59

    1. Initial program 12.5

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

    2. Simplified12.5

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

    if 1.5222805660690795e-59 < b

    1. Initial program 53.0

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

    2. Simplified53.0

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

    4. Applied add-sqr-sqrt53.0

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

    5. Applied sqrt-prod55.2

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

    6. Taylor expanded around inf 18.8

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.367565365770909 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.5222805660690795 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))