Average Error: 33.9 → 10.4
Time: 26.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r120626 = b;
        double r120627 = -r120626;
        double r120628 = r120626 * r120626;
        double r120629 = 3.0;
        double r120630 = a;
        double r120631 = r120629 * r120630;
        double r120632 = c;
        double r120633 = r120631 * r120632;
        double r120634 = r120628 - r120633;
        double r120635 = sqrt(r120634);
        double r120636 = r120627 + r120635;
        double r120637 = r120636 / r120631;
        return r120637;
}
↓
double f(double a, double b, double c) {
        double r120638 = b;
        double r120639 = -1.361733299857302e+105;
        bool r120640 = r120638 <= r120639;
        double r120641 = 0.5;
        double r120642 = c;
        double r120643 = r120642 / r120638;
        double r120644 = r120641 * r120643;
        double r120645 = 0.6666666666666666;
        double r120646 = a;
        double r120647 = r120638 / r120646;
        double r120648 = r120645 * r120647;
        double r120649 = r120644 - r120648;
        double r120650 = 3.091361180800597e-86;
        bool r120651 = r120638 <= r120650;
        double r120652 = r120638 * r120638;
        double r120653 = 3.0;
        double r120654 = r120653 * r120646;
        double r120655 = r120654 * r120642;
        double r120656 = r120652 - r120655;
        double r120657 = sqrt(r120656);
        double r120658 = r120657 - r120638;
        double r120659 = r120658 / r120653;
        double r120660 = r120659 / r120646;
        double r120661 = -0.5;
        double r120662 = r120661 * r120643;
        double r120663 = r120651 ? r120660 : r120662;
        double r120664 = r120640 ? r120649 : r120663;
        return r120664;
}

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 < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    if -1.361733299857302e+105 < b < 3.091361180800597e-86

    1. Initial program 12.2

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

    3. Applied associate-/r*12.3

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

    4. Simplified12.3

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

    if 3.091361180800597e-86 < b

    1. Initial program 51.9

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

    2. Taylor expanded around inf 10.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 3.09136118080059703772253670927164991568 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))