Average Error: 33.3 → 14.8
Time: 34.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3289127594307634 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 7.35606689767469 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -1.3289127594307634 \cdot 10^{+154}:\\
\;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 7.35606689767469 \cdot 10^{-43}:\\
\;\;\;\;\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 r19381522 = b;
        double r19381523 = -r19381522;
        double r19381524 = r19381522 * r19381522;
        double r19381525 = 3.0;
        double r19381526 = a;
        double r19381527 = r19381525 * r19381526;
        double r19381528 = c;
        double r19381529 = r19381527 * r19381528;
        double r19381530 = r19381524 - r19381529;
        double r19381531 = sqrt(r19381530);
        double r19381532 = r19381523 + r19381531;
        double r19381533 = r19381532 / r19381527;
        return r19381533;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r19381534 = b;
        double r19381535 = -1.3289127594307634e+154;
        bool r19381536 = r19381534 <= r19381535;
        double r19381537 = 1.5;
        double r19381538 = a;
        double r19381539 = c;
        double r19381540 = r19381538 * r19381539;
        double r19381541 = r19381540 / r19381534;
        double r19381542 = r19381537 * r19381541;
        double r19381543 = r19381542 - r19381534;
        double r19381544 = r19381543 - r19381534;
        double r19381545 = 3.0;
        double r19381546 = r19381545 * r19381538;
        double r19381547 = r19381544 / r19381546;
        double r19381548 = 7.35606689767469e-43;
        bool r19381549 = r19381534 <= r19381548;
        double r19381550 = r19381534 * r19381534;
        double r19381551 = r19381546 * r19381539;
        double r19381552 = r19381550 - r19381551;
        double r19381553 = sqrt(r19381552);
        double r19381554 = r19381553 - r19381534;
        double r19381555 = r19381554 / r19381546;
        double r19381556 = -1.5;
        double r19381557 = r19381556 * r19381541;
        double r19381558 = r19381557 / r19381546;
        double r19381559 = r19381549 ? r19381555 : r19381558;
        double r19381560 = r19381536 ? r19381547 : r19381559;
        return r19381560;
}

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 < -1.3289127594307634e+154

    1. Initial program 60.8

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

    2. Simplified60.8

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

    3. Taylor expanded around -inf 11.0

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

    if -1.3289127594307634e+154 < b < 7.35606689767469e-43

    1. Initial program 13.2

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

    2. Simplified13.2

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

    if 7.35606689767469e-43 < b

    1. Initial program 54.1

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

    2. Simplified54.1

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

    4. Applied add-cube-cbrt54.1

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

    5. Applied associate-*r*54.1

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

    7. Applied add-cube-cbrt54.1

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

    8. Applied cbrt-prod54.1

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

    9. Taylor expanded around inf 18.1

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3289127594307634 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(\frac{3}{2} \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 7.35606689767469 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))