Average Error: 33.9 → 6.7
Time: 15.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\

\mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r68304 = b;
        double r68305 = -r68304;
        double r68306 = r68304 * r68304;
        double r68307 = 4.0;
        double r68308 = a;
        double r68309 = c;
        double r68310 = r68308 * r68309;
        double r68311 = r68307 * r68310;
        double r68312 = r68306 - r68311;
        double r68313 = sqrt(r68312);
        double r68314 = r68305 - r68313;
        double r68315 = 2.0;
        double r68316 = r68315 * r68308;
        double r68317 = r68314 / r68316;
        return r68317;
}
↓
double f(double a, double b, double c) {
        double r68318 = b;
        double r68319 = -5.674469085146397e+110;
        bool r68320 = r68318 <= r68319;
        double r68321 = -1.0;
        double r68322 = c;
        double r68323 = r68322 / r68318;
        double r68324 = r68321 * r68323;
        double r68325 = 7.061692521831336e-266;
        bool r68326 = r68318 <= r68325;
        double r68327 = 2.0;
        double r68328 = r68327 * r68322;
        double r68329 = 4.0;
        double r68330 = a;
        double r68331 = r68330 * r68322;
        double r68332 = r68329 * r68331;
        double r68333 = -r68332;
        double r68334 = fma(r68318, r68318, r68333);
        double r68335 = sqrt(r68334);
        double r68336 = r68335 - r68318;
        double r68337 = r68328 / r68336;
        double r68338 = 1.7151811081882383e+78;
        bool r68339 = r68318 <= r68338;
        double r68340 = 1.0;
        double r68341 = r68340 / r68327;
        double r68342 = -r68318;
        double r68343 = r68318 * r68318;
        double r68344 = r68343 - r68332;
        double r68345 = sqrt(r68344);
        double r68346 = r68342 - r68345;
        double r68347 = r68346 / r68330;
        double r68348 = r68341 * r68347;
        double r68349 = r68318 / r68330;
        double r68350 = r68321 * r68349;
        double r68351 = r68339 ? r68348 : r68350;
        double r68352 = r68326 ? r68337 : r68351;
        double r68353 = r68320 ? r68324 : r68352;
        return r68353;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.9
Target21.0
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.674469085146397e+110

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.7

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

    if -5.674469085146397e+110 < b < 7.061692521831336e-266

    1. Initial program 31.8

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

    3. Applied flip--31.8

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied div-inv16.1

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]
    8. Using strategy rm

    9. Applied associate-*l/15.2

      \[\leadsto \color{blue}{\frac{\left(0 + \left(a \cdot c\right) \cdot 4\right) \cdot \frac{1}{2 \cdot a}}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}}\]

    10. Simplified15.1

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

    11. Taylor expanded around 0 8.7

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

    if 7.061692521831336e-266 < b < 1.7151811081882383e+78

    1. Initial program 8.6

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

    3. Applied *-un-lft-identity8.6

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

    4. Applied times-frac8.6

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

    if 1.7151811081882383e+78 < b

    1. Initial program 43.0

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

    3. Applied flip--62.6

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    7. Applied div-inv61.8

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}}\]

    8. Taylor expanded around 0 4.8

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 7.061692521831335565675525372535211636164 \cdot 10^{-266}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b}\\ \mathbf{elif}\;b \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

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