Quadratic roots, medium range

Average Error: 43.5 → 11.2

Time: 49.8s

Precision: 64

\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le 0.005377421842056546:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r8378690 = b;
        double r8378691 = -r8378690;
        double r8378692 = r8378690 * r8378690;
        double r8378693 = 4.0;
        double r8378694 = a;
        double r8378695 = r8378693 * r8378694;
        double r8378696 = c;
        double r8378697 = r8378695 * r8378696;
        double r8378698 = r8378692 - r8378697;
        double r8378699 = sqrt(r8378698);
        double r8378700 = r8378691 + r8378699;
        double r8378701 = 2.0;
        double r8378702 = r8378701 * r8378694;
        double r8378703 = r8378700 / r8378702;
        return r8378703;
}

↓

double f(double a, double b, double c) {
        double r8378704 = b;
        double r8378705 = 0.005377421842056546;
        bool r8378706 = r8378704 <= r8378705;
        double r8378707 = r8378704 * r8378704;
        double r8378708 = 4.0;
        double r8378709 = c;
        double r8378710 = a;
        double r8378711 = r8378709 * r8378710;
        double r8378712 = r8378708 * r8378711;
        double r8378713 = r8378707 - r8378712;
        double r8378714 = sqrt(r8378713);
        double r8378715 = r8378713 * r8378714;
        double r8378716 = r8378707 * r8378704;
        double r8378717 = r8378715 - r8378716;
        double r8378718 = 2.0;
        double r8378719 = r8378718 * r8378710;
        double r8378720 = r8378704 * r8378714;
        double r8378721 = r8378707 + r8378720;
        double r8378722 = r8378714 * r8378714;
        double r8378723 = r8378721 + r8378722;
        double r8378724 = r8378719 * r8378723;
        double r8378725 = r8378717 / r8378724;
        double r8378726 = r8378709 / r8378704;
        double r8378727 = -r8378726;
        double r8378728 = r8378706 ? r8378725 : r8378727;
        return r8378728;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 0.005377421842056546

   1. Initial program 19.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Simplified 19.8

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]
   3. Using strategy rm

   4. Applied flip3-- 19.9

      \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{2 \cdot a}\]

   5. Applied associate-/l/ 19.9

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)\right)}}\]

   6. Simplified 19.3

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot \left(b \cdot b\right)}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)\right)}\]

   if 0.005377421842056546 < b

   1. Initial program 46.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Simplified 46.1

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]

   3. Taylor expanded around inf 10.3

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   4. Simplified 10.3

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.005377421842056546:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))