Average Error: 34.2 → 10.3
Time: 33.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.0083428525143135 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 3.780931086400403 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.0083428525143135 \cdot 10^{+138}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 3.780931086400403 \cdot 10^{-21}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot 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 r19083464 = b;
        double r19083465 = -r19083464;
        double r19083466 = r19083464 * r19083464;
        double r19083467 = 3.0;
        double r19083468 = a;
        double r19083469 = r19083467 * r19083468;
        double r19083470 = c;
        double r19083471 = r19083469 * r19083470;
        double r19083472 = r19083466 - r19083471;
        double r19083473 = sqrt(r19083472);
        double r19083474 = r19083465 + r19083473;
        double r19083475 = r19083474 / r19083469;
        return r19083475;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r19083476 = b;
        double r19083477 = -5.0083428525143135e+138;
        bool r19083478 = r19083476 <= r19083477;
        double r19083479 = 0.5;
        double r19083480 = c;
        double r19083481 = r19083480 / r19083476;
        double r19083482 = r19083479 * r19083481;
        double r19083483 = a;
        double r19083484 = r19083476 / r19083483;
        double r19083485 = 0.6666666666666666;
        double r19083486 = r19083484 * r19083485;
        double r19083487 = r19083482 - r19083486;
        double r19083488 = 3.780931086400403e-21;
        bool r19083489 = r19083476 <= r19083488;
        double r19083490 = r19083476 * r19083476;
        double r19083491 = -3.0;
        double r19083492 = r19083480 * r19083491;
        double r19083493 = r19083483 * r19083492;
        double r19083494 = r19083490 + r19083493;
        double r19083495 = sqrt(r19083494);
        double r19083496 = r19083495 - r19083476;
        double r19083497 = 3.0;
        double r19083498 = r19083497 * r19083483;
        double r19083499 = r19083496 / r19083498;
        double r19083500 = -0.5;
        double r19083501 = r19083500 * r19083481;
        double r19083502 = r19083489 ? r19083499 : r19083501;
        double r19083503 = r19083478 ? r19083487 : r19083502;
        return r19083503;
}

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 < -5.0083428525143135e+138

    1. Initial program 55.5

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

    2. Simplified55.5

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

    3. Taylor expanded around 0 55.5

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

    4. Simplified55.5

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

    5. Taylor expanded around -inf 3.7

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

    if -5.0083428525143135e+138 < b < 3.780931086400403e-21

    1. Initial program 14.7

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

    2. Simplified14.7

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

    3. Taylor expanded around 0 14.8

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

    4. Simplified14.8

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

    if 3.780931086400403e-21 < b

    1. Initial program 54.8

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

    2. Simplified54.8

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

    3. Taylor expanded around 0 54.8

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

    4. Simplified54.8

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

    5. Taylor expanded around inf 6.2

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.0083428525143135 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 3.780931086400403 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))