Cubic critical, medium range

Average Error: 43.8 → 11.1

Time: 6.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r82604 = b;
        double r82605 = -r82604;
        double r82606 = r82604 * r82604;
        double r82607 = 3.0;
        double r82608 = a;
        double r82609 = r82607 * r82608;
        double r82610 = c;
        double r82611 = r82609 * r82610;
        double r82612 = r82606 - r82611;
        double r82613 = sqrt(r82612);
        double r82614 = r82605 + r82613;
        double r82615 = r82614 / r82609;
        return r82615;
}

↓

double f(double a, double b, double c) {
        double r82616 = b;
        double r82617 = 0.00017936624356974993;
        bool r82618 = r82616 <= r82617;
        double r82619 = r82616 * r82616;
        double r82620 = c;
        double r82621 = 3.0;
        double r82622 = a;
        double r82623 = r82621 * r82622;
        double r82624 = r82620 * r82623;
        double r82625 = -r82624;
        double r82626 = fma(r82616, r82616, r82625);
        double r82627 = r82619 - r82626;
        double r82628 = -r82616;
        double r82629 = r82623 * r82620;
        double r82630 = r82619 - r82629;
        double r82631 = sqrt(r82630);
        double r82632 = r82628 - r82631;
        double r82633 = r82627 / r82632;
        double r82634 = r82633 / r82623;
        double r82635 = -0.5;
        double r82636 = r82620 / r82616;
        double r82637 = r82635 * r82636;
        double r82638 = r82618 ? r82634 : r82637;
        return r82638;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 0.00017936624356974993

   1. Initial program 18.4

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

   3. Applied flip-+ 18.4

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

   4. Simplified 17.7

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

   if 0.00017936624356974993 < b

   1. Initial program 45.7

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

   2. Taylor expanded around inf 10.6

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.7936624356974993 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, -c \cdot \left(3 \cdot a\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))