Average Error: 33.0 → 9.2
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 8.90802686721313 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 6.105883600684466 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-4, c \cdot a, 0\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{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 8.90802686721313 \cdot 10^{-265}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a}}{2}\\

\mathbf{elif}\;b \le 6.105883600684466 \cdot 10^{+62}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-4, c \cdot a, 0\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2743302 = b;
        double r2743303 = -r2743302;
        double r2743304 = r2743302 * r2743302;
        double r2743305 = 4.0;
        double r2743306 = a;
        double r2743307 = c;
        double r2743308 = r2743306 * r2743307;
        double r2743309 = r2743305 * r2743308;
        double r2743310 = r2743304 - r2743309;
        double r2743311 = sqrt(r2743310);
        double r2743312 = r2743303 + r2743311;
        double r2743313 = 2.0;
        double r2743314 = r2743313 * r2743306;
        double r2743315 = r2743312 / r2743314;
        return r2743315;
}
↓
double f(double a, double b, double c) {
        double r2743316 = b;
        double r2743317 = -9.348931433494438e+39;
        bool r2743318 = r2743316 <= r2743317;
        double r2743319 = c;
        double r2743320 = r2743319 / r2743316;
        double r2743321 = a;
        double r2743322 = r2743316 / r2743321;
        double r2743323 = r2743320 - r2743322;
        double r2743324 = 2.0;
        double r2743325 = r2743323 * r2743324;
        double r2743326 = r2743325 / r2743324;
        double r2743327 = 8.90802686721313e-265;
        bool r2743328 = r2743316 <= r2743327;
        double r2743329 = -4.0;
        double r2743330 = r2743319 * r2743321;
        double r2743331 = r2743329 * r2743330;
        double r2743332 = fma(r2743316, r2743316, r2743331);
        double r2743333 = sqrt(r2743332);
        double r2743334 = r2743333 - r2743316;
        double r2743335 = r2743334 / r2743321;
        double r2743336 = r2743335 / r2743324;
        double r2743337 = 6.105883600684466e+62;
        bool r2743338 = r2743316 <= r2743337;
        double r2743339 = 0.0;
        double r2743340 = fma(r2743329, r2743330, r2743339);
        double r2743341 = r2743340 / r2743321;
        double r2743342 = r2743316 * r2743316;
        double r2743343 = fma(r2743330, r2743329, r2743342);
        double r2743344 = sqrt(r2743343);
        double r2743345 = r2743316 + r2743344;
        double r2743346 = r2743341 / r2743345;
        double r2743347 = r2743346 / r2743324;
        double r2743348 = -2.0;
        double r2743349 = r2743348 * r2743320;
        double r2743350 = r2743349 / r2743324;
        double r2743351 = r2743338 ? r2743347 : r2743350;
        double r2743352 = r2743328 ? r2743336 : r2743351;
        double r2743353 = r2743318 ? r2743326 : r2743352;
        return r2743353;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.1
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    4. Applied div-inv34.1

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

    6. Applied associate-*r/34.0

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

    7. Simplified34.0

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

    8. Taylor expanded around -inf 6.2

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

    9. Simplified6.2

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

    if -9.348931433494438e+39 < b < 8.90802686721313e-265

    1. Initial program 10.1

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

    2. Simplified10.0

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

    4. Applied div-inv10.2

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

    6. Applied associate-*r/10.0

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

    7. Simplified10.0

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

    if 8.90802686721313e-265 < b < 6.105883600684466e+62

    1. Initial program 31.9

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

    2. Simplified31.9

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

    4. Applied div-inv31.9

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

    6. Applied flip--32.0

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

    7. Applied associate-*l/32.1

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

    8. Simplified16.7

      \[\leadsto \frac{\frac{\color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{a}}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{2}\]

    if 6.105883600684466e+62 < b

    1. Initial program 56.7

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

    2. Simplified56.7

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

    4. Applied div-inv56.7

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

    6. Applied associate-*r/56.7

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

    7. Simplified56.7

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

    8. Taylor expanded around inf 3.2

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 8.90802686721313 \cdot 10^{-265}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 6.105883600684466 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(-4, c \cdot a, 0\right)}{a}}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))