Average Error: 33.4 → 9.9
Time: 27.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.0027271082217074 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1555033 = b;
        double r1555034 = -r1555033;
        double r1555035 = r1555033 * r1555033;
        double r1555036 = 4.0;
        double r1555037 = a;
        double r1555038 = r1555036 * r1555037;
        double r1555039 = c;
        double r1555040 = r1555038 * r1555039;
        double r1555041 = r1555035 - r1555040;
        double r1555042 = sqrt(r1555041);
        double r1555043 = r1555034 + r1555042;
        double r1555044 = 2.0;
        double r1555045 = r1555044 * r1555037;
        double r1555046 = r1555043 / r1555045;
        return r1555046;
}
↓
double f(double a, double b, double c) {
        double r1555047 = b;
        double r1555048 = -1.0027271082217074e+110;
        bool r1555049 = r1555047 <= r1555048;
        double r1555050 = c;
        double r1555051 = r1555050 / r1555047;
        double r1555052 = a;
        double r1555053 = r1555047 / r1555052;
        double r1555054 = r1555051 - r1555053;
        double r1555055 = 2.0;
        double r1555056 = r1555054 * r1555055;
        double r1555057 = r1555056 / r1555055;
        double r1555058 = 2.326372645943808e-74;
        bool r1555059 = r1555047 <= r1555058;
        double r1555060 = -4.0;
        double r1555061 = r1555052 * r1555060;
        double r1555062 = r1555047 * r1555047;
        double r1555063 = fma(r1555050, r1555061, r1555062);
        double r1555064 = sqrt(r1555063);
        double r1555065 = r1555064 - r1555047;
        double r1555066 = r1555065 / r1555052;
        double r1555067 = r1555066 / r1555055;
        double r1555068 = -2.0;
        double r1555069 = r1555068 * r1555051;
        double r1555070 = r1555069 / r1555055;
        double r1555071 = r1555059 ? r1555067 : r1555070;
        double r1555072 = r1555049 ? r1555057 : r1555071;
        return r1555072;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0027271082217074e+110

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -1.0027271082217074e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

    4. Simplified12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 8.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))