Average Error: 33.0 → 10.9
Time: 21.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\

\mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2036177 = b;
        double r2036178 = -r2036177;
        double r2036179 = r2036177 * r2036177;
        double r2036180 = 4.0;
        double r2036181 = a;
        double r2036182 = r2036180 * r2036181;
        double r2036183 = c;
        double r2036184 = r2036182 * r2036183;
        double r2036185 = r2036179 - r2036184;
        double r2036186 = sqrt(r2036185);
        double r2036187 = r2036178 + r2036186;
        double r2036188 = 2.0;
        double r2036189 = r2036188 * r2036181;
        double r2036190 = r2036187 / r2036189;
        return r2036190;
}
↓
double f(double a, double b, double c) {
        double r2036191 = b;
        double r2036192 = -9.348931433494438e+39;
        bool r2036193 = r2036191 <= r2036192;
        double r2036194 = c;
        double r2036195 = r2036194 / r2036191;
        double r2036196 = a;
        double r2036197 = r2036191 / r2036196;
        double r2036198 = r2036195 - r2036197;
        double r2036199 = 2.0;
        double r2036200 = r2036198 * r2036199;
        double r2036201 = r2036200 / r2036199;
        double r2036202 = 1.3353078790738604e-121;
        bool r2036203 = r2036191 <= r2036202;
        double r2036204 = 1.0;
        double r2036205 = -4.0;
        double r2036206 = r2036205 * r2036196;
        double r2036207 = r2036194 * r2036206;
        double r2036208 = fma(r2036191, r2036191, r2036207);
        double r2036209 = sqrt(r2036208);
        double r2036210 = r2036209 - r2036191;
        double r2036211 = r2036196 / r2036210;
        double r2036212 = r2036204 / r2036211;
        double r2036213 = r2036212 / r2036199;
        double r2036214 = 1.6168702840263923e-79;
        bool r2036215 = r2036191 <= r2036214;
        double r2036216 = -2.0;
        double r2036217 = r2036195 * r2036216;
        double r2036218 = r2036217 / r2036199;
        double r2036219 = 1.546013236023957e-67;
        bool r2036220 = r2036191 <= r2036219;
        double r2036221 = r2036204 / r2036196;
        double r2036222 = r2036221 * r2036210;
        double r2036223 = r2036222 / r2036199;
        double r2036224 = r2036220 ? r2036223 : r2036218;
        double r2036225 = r2036215 ? r2036218 : r2036224;
        double r2036226 = r2036203 ? r2036213 : r2036225;
        double r2036227 = r2036193 ? r2036201 : r2036226;
        return r2036227;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    4. Applied div-inv34.1

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

    5. Taylor expanded around -inf 6.2

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

    6. Simplified6.2

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

    if -9.348931433494438e+39 < b < 1.3353078790738604e-121

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied clear-num12.3

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

    if 1.3353078790738604e-121 < b < 1.6168702840263923e-79 or 1.546013236023957e-67 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 11.2

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

    if 1.6168702840263923e-79 < b < 1.546013236023957e-67

    1. Initial program 35.8

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

    2. Simplified35.8

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

    4. Applied div-inv35.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right) \cdot \frac{1}{a}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))