Average Error: 28.5 → 16.3
Time: 14.4s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1935.0528928047481:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 1935.0528928047481:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r744624 = b;
        double r744625 = -r744624;
        double r744626 = r744624 * r744624;
        double r744627 = 4.0;
        double r744628 = a;
        double r744629 = r744627 * r744628;
        double r744630 = c;
        double r744631 = r744629 * r744630;
        double r744632 = r744626 - r744631;
        double r744633 = sqrt(r744632);
        double r744634 = r744625 + r744633;
        double r744635 = 2.0;
        double r744636 = r744635 * r744628;
        double r744637 = r744634 / r744636;
        return r744637;
}
↓
double f(double a, double b, double c) {
        double r744638 = b;
        double r744639 = 1935.0528928047481;
        bool r744640 = r744638 <= r744639;
        double r744641 = c;
        double r744642 = -4.0;
        double r744643 = r744641 * r744642;
        double r744644 = a;
        double r744645 = r744638 * r744638;
        double r744646 = fma(r744643, r744644, r744645);
        double r744647 = sqrt(r744646);
        double r744648 = r744647 * r744646;
        double r744649 = r744645 * r744638;
        double r744650 = r744648 - r744649;
        double r744651 = r744645 + r744646;
        double r744652 = fma(r744638, r744647, r744651);
        double r744653 = r744650 / r744652;
        double r744654 = r744653 / r744644;
        double r744655 = 2.0;
        double r744656 = r744654 / r744655;
        double r744657 = -2.0;
        double r744658 = r744641 / r744638;
        double r744659 = r744657 * r744658;
        double r744660 = r744659 / r744655;
        double r744661 = r744640 ? r744656 : r744660;
        return r744661;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 1935.0528928047481

    1. Initial program 17.3

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

    2. Simplified17.3

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

    4. Applied flip3--17.3

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

    5. Simplified16.7

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

    6. Simplified16.7

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

    if 1935.0528928047481 < b

    1. Initial program 36.6

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 16.0

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1935.0528928047481:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b \cdot b + \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(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)))