Average Error: 28.6 → 16.5
Time: 21.4s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot c}{\frac{b}{a} \cdot a}}{2}\\ \end{array}\]
\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 174.55343675894656:\\
\;\;\;\;\frac{\frac{\frac{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b}\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-2 \cdot c}{\frac{b}{a} \cdot a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1411668 = b;
        double r1411669 = -r1411668;
        double r1411670 = r1411668 * r1411668;
        double r1411671 = 4.0;
        double r1411672 = a;
        double r1411673 = r1411671 * r1411672;
        double r1411674 = c;
        double r1411675 = r1411673 * r1411674;
        double r1411676 = r1411670 - r1411675;
        double r1411677 = sqrt(r1411676);
        double r1411678 = r1411669 + r1411677;
        double r1411679 = 2.0;
        double r1411680 = r1411679 * r1411672;
        double r1411681 = r1411678 / r1411680;
        return r1411681;
}
↓
double f(double a, double b, double c) {
        double r1411682 = b;
        double r1411683 = 174.55343675894656;
        bool r1411684 = r1411682 <= r1411683;
        double r1411685 = c;
        double r1411686 = a;
        double r1411687 = r1411685 * r1411686;
        double r1411688 = -4.0;
        double r1411689 = r1411687 * r1411688;
        double r1411690 = r1411682 * r1411682;
        double r1411691 = r1411689 + r1411690;
        double r1411692 = sqrt(r1411691);
        double r1411693 = r1411691 * r1411692;
        double r1411694 = r1411682 * r1411690;
        double r1411695 = r1411693 - r1411694;
        double r1411696 = r1411682 * r1411692;
        double r1411697 = r1411690 + r1411696;
        double r1411698 = r1411691 + r1411697;
        double r1411699 = r1411695 / r1411698;
        double r1411700 = r1411699 / r1411686;
        double r1411701 = 2.0;
        double r1411702 = r1411700 / r1411701;
        double r1411703 = -2.0;
        double r1411704 = r1411703 * r1411685;
        double r1411705 = r1411682 / r1411686;
        double r1411706 = r1411705 * r1411686;
        double r1411707 = r1411704 / r1411706;
        double r1411708 = r1411707 / r1411701;
        double r1411709 = r1411684 ? r1411702 : r1411708;
        return r1411709;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 174.55343675894656

    1. Initial program 15.7

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

    2. Simplified15.7

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

    4. Applied flip3--15.8

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

    5. Simplified15.1

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

    6. Simplified15.0

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

    if 174.55343675894656 < b

    1. Initial program 35.0

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

    2. Simplified35.0

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

    3. Taylor expanded around inf 17.3

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a}}{2}\]
    4. Using strategy rm

    5. Applied *-commutative17.3

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

    6. Applied associate-/l*17.2

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

    7. Applied associate-*r/17.2

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

    8. Applied associate-/l/17.2

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot c}{\frac{b}{a} \cdot a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))