Average Error: 28.3 → 16.3
Time: 7.1s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 904.955392750566375:\\
\;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r22095 = b;
        double r22096 = -r22095;
        double r22097 = r22095 * r22095;
        double r22098 = 4.0;
        double r22099 = a;
        double r22100 = r22098 * r22099;
        double r22101 = c;
        double r22102 = r22100 * r22101;
        double r22103 = r22097 - r22102;
        double r22104 = sqrt(r22103);
        double r22105 = r22096 + r22104;
        double r22106 = 2.0;
        double r22107 = r22106 * r22099;
        double r22108 = r22105 / r22107;
        return r22108;
}
↓
double f(double a, double b, double c) {
        double r22109 = b;
        double r22110 = 904.9553927505664;
        bool r22111 = r22109 <= r22110;
        double r22112 = 2.0;
        double r22113 = pow(r22109, r22112);
        double r22114 = 4.0;
        double r22115 = a;
        double r22116 = c;
        double r22117 = r22115 * r22116;
        double r22118 = r22114 * r22117;
        double r22119 = r22113 - r22118;
        double r22120 = r22119 - r22113;
        double r22121 = r22109 * r22109;
        double r22122 = r22114 * r22115;
        double r22123 = r22122 * r22116;
        double r22124 = r22121 - r22123;
        double r22125 = sqrt(r22124);
        double r22126 = r22125 + r22109;
        double r22127 = r22120 / r22126;
        double r22128 = 2.0;
        double r22129 = r22127 / r22128;
        double r22130 = r22129 / r22115;
        double r22131 = -1.0;
        double r22132 = r22116 / r22109;
        double r22133 = r22131 * r22132;
        double r22134 = r22111 ? r22130 : r22133;
        return r22134;
}

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 < 904.9553927505664

    1. Initial program 17.1

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

    2. Simplified17.1

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

    4. Applied flip--17.1

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

    5. Simplified16.0

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

    if 904.9553927505664 < b

    1. Initial program 36.0

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 16.5

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))