Average Error: 28.3 → 16.8
Time: 17.7s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 60.51244836482759836826517130248248577118:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 60.51244836482759836826517130248248577118:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1698469 = b;
        double r1698470 = -r1698469;
        double r1698471 = r1698469 * r1698469;
        double r1698472 = 4.0;
        double r1698473 = a;
        double r1698474 = r1698472 * r1698473;
        double r1698475 = c;
        double r1698476 = r1698474 * r1698475;
        double r1698477 = r1698471 - r1698476;
        double r1698478 = sqrt(r1698477);
        double r1698479 = r1698470 + r1698478;
        double r1698480 = 2.0;
        double r1698481 = r1698480 * r1698473;
        double r1698482 = r1698479 / r1698481;
        return r1698482;
}
↓
double f(double a, double b, double c) {
        double r1698483 = b;
        double r1698484 = 60.5124483648276;
        bool r1698485 = r1698483 <= r1698484;
        double r1698486 = r1698483 * r1698483;
        double r1698487 = a;
        double r1698488 = c;
        double r1698489 = r1698487 * r1698488;
        double r1698490 = 4.0;
        double r1698491 = r1698489 * r1698490;
        double r1698492 = r1698486 - r1698491;
        double r1698493 = sqrt(r1698492);
        double r1698494 = r1698492 * r1698493;
        double r1698495 = r1698486 * r1698483;
        double r1698496 = r1698494 - r1698495;
        double r1698497 = fma(r1698493, r1698483, r1698492);
        double r1698498 = r1698497 + r1698486;
        double r1698499 = r1698496 / r1698498;
        double r1698500 = r1698499 / r1698487;
        double r1698501 = 2.0;
        double r1698502 = r1698500 / r1698501;
        double r1698503 = -2.0;
        double r1698504 = r1698488 / r1698483;
        double r1698505 = r1698503 * r1698504;
        double r1698506 = r1698505 / r1698501;
        double r1698507 = r1698485 ? r1698502 : r1698506;
        return r1698507;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 60.5124483648276

    1. Initial program 14.3

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

    2. Simplified14.3

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

    4. Applied flip3--14.4

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

    5. Simplified13.7

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

    6. Simplified13.7

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

    if 60.5124483648276 < b

    1. Initial program 33.9

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

    2. Simplified33.9

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

    3. Taylor expanded around inf 18.1

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 60.51244836482759836826517130248248577118:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}, b, b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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) (* (* 4.0 a) c)))) (* 2.0 a)))