Average Error: 33.7 → 10.9
Time: 10.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.98276540088900058 \cdot 10^{134}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\
\;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r56050 = b;
        double r56051 = -r56050;
        double r56052 = r56050 * r56050;
        double r56053 = 4.0;
        double r56054 = a;
        double r56055 = r56053 * r56054;
        double r56056 = c;
        double r56057 = r56055 * r56056;
        double r56058 = r56052 - r56057;
        double r56059 = sqrt(r56058);
        double r56060 = r56051 + r56059;
        double r56061 = 2.0;
        double r56062 = r56061 * r56054;
        double r56063 = r56060 / r56062;
        return r56063;
}
↓
double f(double a, double b, double c) {
        double r56064 = b;
        double r56065 = -1.9827654008890006e+134;
        bool r56066 = r56064 <= r56065;
        double r56067 = 1.0;
        double r56068 = c;
        double r56069 = r56068 / r56064;
        double r56070 = a;
        double r56071 = r56064 / r56070;
        double r56072 = r56069 - r56071;
        double r56073 = r56067 * r56072;
        double r56074 = 1.1860189201379418e-161;
        bool r56075 = r56064 <= r56074;
        double r56076 = 4.0;
        double r56077 = r56076 * r56070;
        double r56078 = r56077 * r56068;
        double r56079 = -r56078;
        double r56080 = fma(r56064, r56064, r56079);
        double r56081 = sqrt(r56080);
        double r56082 = r56081 - r56064;
        double r56083 = 1.0;
        double r56084 = 2.0;
        double r56085 = r56083 / r56084;
        double r56086 = r56085 / r56070;
        double r56087 = r56082 * r56086;
        double r56088 = -1.0;
        double r56089 = r56088 * r56069;
        double r56090 = r56075 ? r56087 : r56089;
        double r56091 = r56066 ? r56073 : r56090;
        return r56091;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.9827654008890006e+134

    1. Initial program 56.8

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

    2. Simplified56.8

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

    3. Taylor expanded around -inf 3.1

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

    4. Simplified3.1

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

    if -1.9827654008890006e+134 < b < 1.1860189201379418e-161

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied *-un-lft-identity10.3

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

    5. Applied div-inv10.3

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

    6. Applied times-frac10.5

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

    7. Simplified10.5

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

    if 1.1860189201379418e-161 < b

    1. Initial program 49.7

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

    2. Simplified49.7

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

    3. Taylor expanded around inf 13.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.98276540088900058 \cdot 10^{134}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, -\left(4 \cdot a\right) \cdot c\right)} - b\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))