Average Error: 33.4 → 15.1
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3532058622703094 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 5.9471276972445346 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]
\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 -1.3532058622703094 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\

\mathbf{elif}\;b \le 5.9471276972445346 \cdot 10^{-61}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4600561 = b;
        double r4600562 = -r4600561;
        double r4600563 = r4600561 * r4600561;
        double r4600564 = 3.0;
        double r4600565 = a;
        double r4600566 = r4600564 * r4600565;
        double r4600567 = c;
        double r4600568 = r4600566 * r4600567;
        double r4600569 = r4600563 - r4600568;
        double r4600570 = sqrt(r4600569);
        double r4600571 = r4600562 + r4600570;
        double r4600572 = r4600571 / r4600566;
        return r4600572;
}
↓
double f(double a, double b, double c) {
        double r4600573 = b;
        double r4600574 = -1.3532058622703094e+154;
        bool r4600575 = r4600573 <= r4600574;
        double r4600576 = 1.5;
        double r4600577 = a;
        double r4600578 = c;
        double r4600579 = r4600577 * r4600578;
        double r4600580 = r4600579 / r4600573;
        double r4600581 = r4600576 * r4600580;
        double r4600582 = 2.0;
        double r4600583 = r4600582 * r4600573;
        double r4600584 = r4600581 - r4600583;
        double r4600585 = 3.0;
        double r4600586 = r4600577 * r4600585;
        double r4600587 = r4600584 / r4600586;
        double r4600588 = 5.9471276972445346e-61;
        bool r4600589 = r4600573 <= r4600588;
        double r4600590 = r4600573 * r4600573;
        double r4600591 = r4600585 * r4600578;
        double r4600592 = r4600577 * r4600591;
        double r4600593 = r4600590 - r4600592;
        double r4600594 = sqrt(r4600593);
        double r4600595 = r4600594 - r4600573;
        double r4600596 = r4600595 / r4600586;
        double r4600597 = -1.5;
        double r4600598 = r4600597 * r4600580;
        double r4600599 = r4600598 / r4600586;
        double r4600600 = r4600589 ? r4600596 : r4600599;
        double r4600601 = r4600575 ? r4600587 : r4600600;
        return r4600601;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3532058622703094e+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 61.0

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

    5. Applied add-cube-cbrt61.0

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

    6. Applied associate-*l*61.0

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

    7. Taylor expanded around -inf 11.8

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

    if -1.3532058622703094e+154 < b < 5.9471276972445346e-61

    1. Initial program 12.6

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

    2. Simplified12.6

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

    3. Taylor expanded around inf 12.7

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

    5. Applied add-cube-cbrt12.7

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

    6. Applied associate-*l*12.7

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

    7. Taylor expanded around -inf 12.7

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

    8. Taylor expanded around 0 12.7

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

    9. Simplified12.7

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

    if 5.9471276972445346e-61 < b

    1. Initial program 52.6

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

    2. Simplified52.6

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

    3. Taylor expanded around inf 52.6

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

    4. 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 simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3532058622703094 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3}{2} \cdot \frac{a \cdot c}{b} - 2 \cdot b}{a \cdot 3}\\ \mathbf{elif}\;b \le 5.9471276972445346 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))