Average Error: 28.4 → 16.4
Time: 22.5s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 835.2343651472419878700748085975646972656:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}, b \cdot b\right) + \left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}}{2}}{\sqrt{a}} \cdot \frac{-2}{\sqrt{a}}\\ \end{array}\]
\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 835.2343651472419878700748085975646972656:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}, b \cdot b\right) + \left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1956503 = b;
        double r1956504 = -r1956503;
        double r1956505 = r1956503 * r1956503;
        double r1956506 = 4.0;
        double r1956507 = a;
        double r1956508 = r1956506 * r1956507;
        double r1956509 = c;
        double r1956510 = r1956508 * r1956509;
        double r1956511 = r1956505 - r1956510;
        double r1956512 = sqrt(r1956511);
        double r1956513 = r1956504 + r1956512;
        double r1956514 = 2.0;
        double r1956515 = r1956514 * r1956507;
        double r1956516 = r1956513 / r1956515;
        return r1956516;
}
↓
double f(double a, double b, double c) {
        double r1956517 = b;
        double r1956518 = 835.234365147242;
        bool r1956519 = r1956517 <= r1956518;
        double r1956520 = r1956517 * r1956517;
        double r1956521 = 4.0;
        double r1956522 = c;
        double r1956523 = a;
        double r1956524 = r1956522 * r1956523;
        double r1956525 = r1956521 * r1956524;
        double r1956526 = r1956520 - r1956525;
        double r1956527 = sqrt(r1956526);
        double r1956528 = r1956526 * r1956527;
        double r1956529 = r1956520 * r1956517;
        double r1956530 = r1956528 - r1956529;
        double r1956531 = fma(r1956517, r1956527, r1956520);
        double r1956532 = r1956531 + r1956526;
        double r1956533 = r1956530 / r1956532;
        double r1956534 = 2.0;
        double r1956535 = r1956533 / r1956534;
        double r1956536 = r1956535 / r1956523;
        double r1956537 = sqrt(r1956517);
        double r1956538 = r1956523 / r1956537;
        double r1956539 = r1956522 / r1956537;
        double r1956540 = r1956538 * r1956539;
        double r1956541 = r1956540 / r1956534;
        double r1956542 = sqrt(r1956523);
        double r1956543 = r1956541 / r1956542;
        double r1956544 = -2.0;
        double r1956545 = r1956544 / r1956542;
        double r1956546 = r1956543 * r1956545;
        double r1956547 = r1956519 ? r1956536 : r1956546;
        return r1956547;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 835.234365147242

    1. Initial program 16.7

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

    2. Simplified16.7

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

    4. Applied flip3--16.8

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

    5. Simplified16.1

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

    6. Simplified16.1

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

    if 835.234365147242 < b

    1. Initial program 36.1

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 16.5

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

    5. Applied add-sqr-sqrt16.6

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

    6. Applied times-frac16.6

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

    8. Applied add-sqr-sqrt16.6

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

    9. Applied *-un-lft-identity16.6

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

    10. Applied times-frac16.6

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

    11. Applied times-frac16.6

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

    12. Simplified16.6

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 835.2343651472419878700748085975646972656:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}, b \cdot b\right) + \left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}}{2}}{\sqrt{a}} \cdot \frac{-2}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))