Average Error: 33.4 → 16.7
Time: 25.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 8.670930634061063 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]
\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 8.670930634061063 \cdot 10^{-143}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1310885 = b;
        double r1310886 = -r1310885;
        double r1310887 = r1310885 * r1310885;
        double r1310888 = 4.0;
        double r1310889 = a;
        double r1310890 = r1310888 * r1310889;
        double r1310891 = c;
        double r1310892 = r1310890 * r1310891;
        double r1310893 = r1310887 - r1310892;
        double r1310894 = sqrt(r1310893);
        double r1310895 = r1310886 + r1310894;
        double r1310896 = 2.0;
        double r1310897 = r1310896 * r1310889;
        double r1310898 = r1310895 / r1310897;
        return r1310898;
}
↓
double f(double a, double b, double c) {
        double r1310899 = b;
        double r1310900 = 8.670930634061063e-143;
        bool r1310901 = r1310899 <= r1310900;
        double r1310902 = c;
        double r1310903 = a;
        double r1310904 = -4.0;
        double r1310905 = r1310903 * r1310904;
        double r1310906 = r1310899 * r1310899;
        double r1310907 = fma(r1310902, r1310905, r1310906);
        double r1310908 = sqrt(r1310907);
        double r1310909 = r1310908 - r1310899;
        double r1310910 = 0.5;
        double r1310911 = r1310909 * r1310910;
        double r1310912 = r1310911 / r1310903;
        double r1310913 = 1.0;
        double r1310914 = r1310899 / r1310902;
        double r1310915 = -r1310914;
        double r1310916 = r1310913 / r1310915;
        double r1310917 = r1310901 ? r1310912 : r1310916;
        return r1310917;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 8.670930634061063e-143

    1. Initial program 20.2

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

    2. Simplified20.2

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

    4. Applied *-un-lft-identity20.2

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b}{2}}{\color{blue}{1 \cdot a}}\]

    5. Applied div-inv20.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{2}}}{1 \cdot a}\]

    6. Applied times-frac20.3

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b}{1} \cdot \frac{\frac{1}{2}}{a}}\]

    7. Simplified20.3

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right)} \cdot \frac{\frac{1}{2}}{a}\]

    8. Simplified20.3

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
    9. Using strategy rm

    10. Applied associate-*r/20.2

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{2}}{a}}\]

    if 8.670930634061063e-143 < b

    1. Initial program 49.8

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

    2. Simplified49.8

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

    4. Applied *-un-lft-identity49.8

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b}{2}}}{a}\]

    5. Applied associate-/l*49.9

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b}{2}}}}\]

    6. Taylor expanded around 0 12.3

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

    7. Simplified12.3

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 8.670930634061063 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))