Cubic critical, wide range

Average Error: 52.5 → 5.8

Time: 24.8s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3} \le -117792.8831713972322177141904830932617188:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(c \cdot 3\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b \cdot \left(b \cdot b\right)}{b \cdot b + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} + \left(b \cdot b - a \cdot \left(c \cdot 3\right)\right)\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r4591621 = b;
        double r4591622 = -r4591621;
        double r4591623 = r4591621 * r4591621;
        double r4591624 = 3.0;
        double r4591625 = a;
        double r4591626 = r4591624 * r4591625;
        double r4591627 = c;
        double r4591628 = r4591626 * r4591627;
        double r4591629 = r4591623 - r4591628;
        double r4591630 = sqrt(r4591629);
        double r4591631 = r4591622 + r4591630;
        double r4591632 = r4591631 / r4591626;
        return r4591632;
}

↓

double f(double a, double b, double c) {
        double r4591633 = b;
        double r4591634 = -r4591633;
        double r4591635 = r4591633 * r4591633;
        double r4591636 = a;
        double r4591637 = 3.0;
        double r4591638 = r4591636 * r4591637;
        double r4591639 = c;
        double r4591640 = r4591638 * r4591639;
        double r4591641 = r4591635 - r4591640;
        double r4591642 = sqrt(r4591641);
        double r4591643 = r4591634 + r4591642;
        double r4591644 = r4591643 / r4591638;
        double r4591645 = -117792.88317139723;
        bool r4591646 = r4591644 <= r4591645;
        double r4591647 = r4591639 * r4591637;
        double r4591648 = r4591636 * r4591647;
        double r4591649 = r4591635 - r4591648;
        double r4591650 = sqrt(r4591649);
        double r4591651 = r4591649 * r4591650;
        double r4591652 = r4591633 * r4591635;
        double r4591653 = r4591651 - r4591652;
        double r4591654 = r4591633 * r4591650;
        double r4591655 = r4591654 + r4591649;
        double r4591656 = r4591635 + r4591655;
        double r4591657 = r4591653 / r4591656;
        double r4591658 = r4591657 / r4591638;
        double r4591659 = -0.5;
        double r4591660 = r4591639 / r4591633;
        double r4591661 = r4591659 * r4591660;
        double r4591662 = r4591646 ? r4591658 : r4591661;
        return r4591662;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) < -117792.88317139723

   1. Initial program 21.9

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

   3. Applied flip3-+ 22.0

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

   4. Simplified 21.3

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

   5. Simplified 21.3

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

   if -117792.88317139723 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

   1. Initial program 54.7

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

   2. Taylor expanded around inf 4.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{a \cdot 3} \le -117792.8831713972322177141904830932617188:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(c \cdot 3\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b \cdot \left(b \cdot b\right)}{b \cdot b + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} + \left(b \cdot b - a \cdot \left(c \cdot 3\right)\right)\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))