Average Error: 34.0 → 15.4
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.357880986558193 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.8449131242087044 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.357880986558193 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 2.8449131242087044 \cdot 10^{-58}:\\
\;\;\;\;\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 r2638482 = b;
        double r2638483 = -r2638482;
        double r2638484 = r2638482 * r2638482;
        double r2638485 = 3.0;
        double r2638486 = a;
        double r2638487 = r2638485 * r2638486;
        double r2638488 = c;
        double r2638489 = r2638487 * r2638488;
        double r2638490 = r2638484 - r2638489;
        double r2638491 = sqrt(r2638490);
        double r2638492 = r2638483 + r2638491;
        double r2638493 = r2638492 / r2638487;
        return r2638493;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r2638494 = b;
        double r2638495 = -1.357880986558193e+154;
        bool r2638496 = r2638494 <= r2638495;
        double r2638497 = 1.5;
        double r2638498 = a;
        double r2638499 = c;
        double r2638500 = r2638498 * r2638499;
        double r2638501 = r2638500 / r2638494;
        double r2638502 = r2638497 * r2638501;
        double r2638503 = r2638502 - r2638494;
        double r2638504 = r2638503 - r2638494;
        double r2638505 = 3.0;
        double r2638506 = r2638505 * r2638498;
        double r2638507 = r2638504 / r2638506;
        double r2638508 = 2.8449131242087044e-58;
        bool r2638509 = r2638494 <= r2638508;
        double r2638510 = r2638494 * r2638494;
        double r2638511 = r2638506 * r2638499;
        double r2638512 = r2638510 - r2638511;
        double r2638513 = sqrt(r2638512);
        double r2638514 = r2638513 - r2638494;
        double r2638515 = r2638514 / r2638506;
        double r2638516 = -1.5;
        double r2638517 = r2638516 * r2638501;
        double r2638518 = r2638517 / r2638506;
        double r2638519 = r2638509 ? r2638515 : r2638518;
        double r2638520 = r2638496 ? r2638507 : r2638519;
        return r2638520;
}

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 < -1.357880986558193e+154

    1. Initial program 60.9

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

    2. Simplified60.9

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

    3. Taylor expanded around -inf 10.8

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

    if -1.357880986558193e+154 < b < 2.8449131242087044e-58

    1. Initial program 13.5

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

    2. Simplified13.5

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

    if 2.8449131242087044e-58 < b

    1. Initial program 53.7

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

    2. Simplified53.7

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

    4. Applied add-sqr-sqrt53.7

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

    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 19.2

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

  4. Final simplification15.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.357880986558193 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.8449131242087044 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))