Average Error: 34.9 → 10.2
Time: 16.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.386028720359615261249042939934860150641 \cdot 10^{148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 1.5, b \cdot -2\right)}{3}}{a}\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.386028720359615261249042939934860150641 \cdot 10^{148}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 1.5, b \cdot -2\right)}{3}}{a}\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4470520 = b;
        double r4470521 = -r4470520;
        double r4470522 = r4470520 * r4470520;
        double r4470523 = 3.0;
        double r4470524 = a;
        double r4470525 = r4470523 * r4470524;
        double r4470526 = c;
        double r4470527 = r4470525 * r4470526;
        double r4470528 = r4470522 - r4470527;
        double r4470529 = sqrt(r4470528);
        double r4470530 = r4470521 + r4470529;
        double r4470531 = r4470530 / r4470525;
        return r4470531;
}
↓
double f(double a, double b, double c) {
        double r4470532 = b;
        double r4470533 = -6.386028720359615e+148;
        bool r4470534 = r4470532 <= r4470533;
        double r4470535 = c;
        double r4470536 = a;
        double r4470537 = r4470532 / r4470536;
        double r4470538 = r4470535 / r4470537;
        double r4470539 = 1.5;
        double r4470540 = -2.0;
        double r4470541 = r4470532 * r4470540;
        double r4470542 = fma(r4470538, r4470539, r4470541);
        double r4470543 = 3.0;
        double r4470544 = r4470542 / r4470543;
        double r4470545 = r4470544 / r4470536;
        double r4470546 = 4.6129908231112306e-104;
        bool r4470547 = r4470532 <= r4470546;
        double r4470548 = 1.0;
        double r4470549 = r4470543 * r4470536;
        double r4470550 = r4470532 * r4470532;
        double r4470551 = r4470535 * r4470549;
        double r4470552 = r4470550 - r4470551;
        double r4470553 = sqrt(r4470552);
        double r4470554 = r4470553 - r4470532;
        double r4470555 = r4470549 / r4470554;
        double r4470556 = r4470548 / r4470555;
        double r4470557 = -0.5;
        double r4470558 = r4470535 / r4470532;
        double r4470559 = r4470557 * r4470558;
        double r4470560 = r4470547 ? r4470556 : r4470559;
        double r4470561 = r4470534 ? r4470545 : r4470560;
        return r4470561;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -6.386028720359615e+148

    1. Initial program 62.3

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

    3. Applied associate-/r*62.3

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

    4. Simplified62.3

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

    5. Taylor expanded around -inf 11.6

      \[\leadsto \frac{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified2.7

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 1.5, b \cdot -2\right)}}{3}}{a}\]

    if -6.386028720359615e+148 < b < 4.6129908231112306e-104

    1. Initial program 12.3

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

    3. Applied clear-num12.3

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}\]

    4. Simplified12.3

      \[\leadsto \frac{1}{\color{blue}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}}}\]

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Taylor expanded around inf 9.8

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.386028720359615261249042939934860150641 \cdot 10^{148}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 1.5, b \cdot -2\right)}{3}}{a}\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))