Average Error: 28.5 → 16.6
Time: 15.9s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5152.464935290847:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(c \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot -2}{a}}{2}\\ \end{array}\]
\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 5152.464935290847:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\left(\left(c \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot -2}{a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1375112 = b;
        double r1375113 = -r1375112;
        double r1375114 = r1375112 * r1375112;
        double r1375115 = 4.0;
        double r1375116 = a;
        double r1375117 = r1375115 * r1375116;
        double r1375118 = c;
        double r1375119 = r1375117 * r1375118;
        double r1375120 = r1375114 - r1375119;
        double r1375121 = sqrt(r1375120);
        double r1375122 = r1375113 + r1375121;
        double r1375123 = 2.0;
        double r1375124 = r1375123 * r1375116;
        double r1375125 = r1375122 / r1375124;
        return r1375125;
}
↓
double f(double a, double b, double c) {
        double r1375126 = b;
        double r1375127 = 5152.464935290847;
        bool r1375128 = r1375126 <= r1375127;
        double r1375129 = -4.0;
        double r1375130 = a;
        double r1375131 = r1375129 * r1375130;
        double r1375132 = c;
        double r1375133 = r1375126 * r1375126;
        double r1375134 = fma(r1375131, r1375132, r1375133);
        double r1375135 = sqrt(r1375134);
        double r1375136 = r1375135 * r1375134;
        double r1375137 = r1375133 * r1375126;
        double r1375138 = r1375136 - r1375137;
        double r1375139 = r1375126 + r1375135;
        double r1375140 = fma(r1375126, r1375139, r1375134);
        double r1375141 = r1375138 / r1375140;
        double r1375142 = r1375141 / r1375130;
        double r1375143 = 2.0;
        double r1375144 = r1375142 / r1375143;
        double r1375145 = sqrt(r1375130);
        double r1375146 = sqrt(r1375126);
        double r1375147 = r1375145 / r1375146;
        double r1375148 = r1375132 * r1375147;
        double r1375149 = r1375148 * r1375147;
        double r1375150 = -2.0;
        double r1375151 = r1375149 * r1375150;
        double r1375152 = r1375151 / r1375130;
        double r1375153 = r1375152 / r1375143;
        double r1375154 = r1375128 ? r1375144 : r1375153;
        return r1375154;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 5152.464935290847

    1. Initial program 18.7

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

    2. Simplified18.6

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

    4. Applied flip3--18.8

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

    5. Simplified18.1

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

    6. Simplified18.1

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

    if 5152.464935290847 < b

    1. Initial program 37.9

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

    2. Simplified37.8

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

    3. Taylor expanded around inf 15.1

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

    5. Applied associate-/l*15.1

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

    7. Applied *-un-lft-identity15.1

      \[\leadsto \frac{\frac{-2 \cdot \frac{a}{\frac{b}{\color{blue}{1 \cdot c}}}}{a}}{2}\]

    8. Applied add-sqr-sqrt15.2

      \[\leadsto \frac{\frac{-2 \cdot \frac{a}{\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}{1 \cdot c}}}{a}}{2}\]

    9. Applied times-frac15.2

      \[\leadsto \frac{\frac{-2 \cdot \frac{a}{\color{blue}{\frac{\sqrt{b}}{1} \cdot \frac{\sqrt{b}}{c}}}}{a}}{2}\]

    10. Applied add-sqr-sqrt15.2

      \[\leadsto \frac{\frac{-2 \cdot \frac{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}{\frac{\sqrt{b}}{1} \cdot \frac{\sqrt{b}}{c}}}{a}}{2}\]

    11. Applied times-frac15.2

      \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\left(\frac{\sqrt{a}}{\frac{\sqrt{b}}{1}} \cdot \frac{\sqrt{a}}{\frac{\sqrt{b}}{c}}\right)}}{a}}{2}\]

    12. Simplified15.2

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

    13. Simplified15.2

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5152.464935290847:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(c \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot \frac{\sqrt{a}}{\sqrt{b}}\right) \cdot -2}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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)))