Cubic critical, narrow range

Average Error: 28.6 → 16.6

Time: 46.5s

Precision: 64

• unused variable d

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 92.71167522702159:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15070604 = b;
        double r15070605 = -r15070604;
        double r15070606 = r15070604 * r15070604;
        double r15070607 = 3.0;
        double r15070608 = a;
        double r15070609 = r15070607 * r15070608;
        double r15070610 = c;
        double r15070611 = r15070609 * r15070610;
        double r15070612 = r15070606 - r15070611;
        double r15070613 = sqrt(r15070612);
        double r15070614 = r15070605 + r15070613;
        double r15070615 = r15070614 / r15070609;
        return r15070615;
}

↓

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r15070616 = b;
        double r15070617 = 92.71167522702159;
        bool r15070618 = r15070616 <= r15070617;
        double r15070619 = r15070616 * r15070616;
        double r15070620 = c;
        double r15070621 = 3.0;
        double r15070622 = a;
        double r15070623 = r15070621 * r15070622;
        double r15070624 = r15070620 * r15070623;
        double r15070625 = r15070619 - r15070624;
        double r15070626 = sqrt(r15070625);
        double r15070627 = r15070625 * r15070626;
        double r15070628 = r15070619 * r15070616;
        double r15070629 = r15070627 - r15070628;
        double r15070630 = r15070616 * r15070626;
        double r15070631 = r15070630 + r15070619;
        double r15070632 = r15070626 * r15070626;
        double r15070633 = r15070631 + r15070632;
        double r15070634 = r15070623 * r15070633;
        double r15070635 = r15070629 / r15070634;
        double r15070636 = -0.5;
        double r15070637 = r15070620 / r15070616;
        double r15070638 = r15070636 * r15070637;
        double r15070639 = r15070618 ? r15070635 : r15070638;
        return r15070639;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

1. Split input into 2 regimes

2. if b < 92.71167522702159

   1. Initial program 15.1

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

   2. Simplified 15.1

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

   4. Applied flip3-- 15.2

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

   5. Applied associate-/l/ 15.2

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

   6. Simplified 14.5

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

   if 92.71167522702159 < b

   1. Initial program 34.6

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

   2. Simplified 34.6

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

   3. Taylor expanded around inf 17.6

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 92.71167522702159:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))