Average Error: 28.3 → 16.7
Time: 12.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 434.5586936141062892602349165827035903931:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 434.5586936141062892602349165827035903931:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r26691 = b;
        double r26692 = -r26691;
        double r26693 = r26691 * r26691;
        double r26694 = 4.0;
        double r26695 = a;
        double r26696 = r26694 * r26695;
        double r26697 = c;
        double r26698 = r26696 * r26697;
        double r26699 = r26693 - r26698;
        double r26700 = sqrt(r26699);
        double r26701 = r26692 + r26700;
        double r26702 = 2.0;
        double r26703 = r26702 * r26695;
        double r26704 = r26701 / r26703;
        return r26704;
}
↓
double f(double a, double b, double c) {
        double r26705 = b;
        double r26706 = 434.5586936141063;
        bool r26707 = r26705 <= r26706;
        double r26708 = r26705 * r26705;
        double r26709 = 4.0;
        double r26710 = a;
        double r26711 = c;
        double r26712 = r26710 * r26711;
        double r26713 = r26709 * r26712;
        double r26714 = fma(r26705, r26705, r26713);
        double r26715 = r26708 - r26714;
        double r26716 = r26709 * r26710;
        double r26717 = r26716 * r26711;
        double r26718 = r26708 - r26717;
        double r26719 = sqrt(r26718);
        double r26720 = r26719 + r26705;
        double r26721 = r26715 / r26720;
        double r26722 = 2.0;
        double r26723 = r26722 * r26710;
        double r26724 = r26721 / r26723;
        double r26725 = -1.0;
        double r26726 = r26711 / r26705;
        double r26727 = r26725 * r26726;
        double r26728 = r26707 ? r26724 : r26727;
        return r26728;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 434.5586936141063

    1. Initial program 16.4

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

    2. Simplified16.4

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

    4. Applied flip--16.4

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

    5. Simplified15.6

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

    if 434.5586936141063 < 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

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

    3. Taylor expanded around inf 17.3

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 434.5586936141062892602349165827035903931:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, 4 \cdot \left(a \cdot c\right)\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :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)))