Average Error: 32.9 → 10.3
Time: 20.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r724078 = b;
        double r724079 = -r724078;
        double r724080 = r724078 * r724078;
        double r724081 = 4.0;
        double r724082 = a;
        double r724083 = r724081 * r724082;
        double r724084 = c;
        double r724085 = r724083 * r724084;
        double r724086 = r724080 - r724085;
        double r724087 = sqrt(r724086);
        double r724088 = r724079 + r724087;
        double r724089 = 2.0;
        double r724090 = r724089 * r724082;
        double r724091 = r724088 / r724090;
        return r724091;
}
↓
double f(double a, double b, double c) {
        double r724092 = b;
        double r724093 = -9.088000531423294e+152;
        bool r724094 = r724092 <= r724093;
        double r724095 = c;
        double r724096 = r724095 / r724092;
        double r724097 = a;
        double r724098 = r724092 / r724097;
        double r724099 = r724096 - r724098;
        double r724100 = 9.354082991670835e-125;
        bool r724101 = r724092 <= r724100;
        double r724102 = -4.0;
        double r724103 = r724097 * r724102;
        double r724104 = r724092 * r724092;
        double r724105 = fma(r724103, r724095, r724104);
        double r724106 = sqrt(r724105);
        double r724107 = r724106 - r724092;
        double r724108 = 0.5;
        double r724109 = r724107 * r724108;
        double r724110 = r724109 / r724097;
        double r724111 = -r724095;
        double r724112 = r724111 / r724092;
        double r724113 = r724101 ? r724110 : r724112;
        double r724114 = r724094 ? r724099 : r724113;
        return r724114;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -9.088000531423294e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied *-un-lft-identity10.9

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

    5. Applied div-inv11.1

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

    6. Applied times-frac11.1

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

    7. Simplified11.1

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

    8. Simplified11.1

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

    10. Applied associate-*r/10.9

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

    if 9.354082991670835e-125 < 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

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

    3. Taylor expanded around inf 11.9

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

    4. Simplified11.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))