Average Error: 43.7 → 11.6
Time: 51.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.003195690106913911:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{elif}\;b \le 0.1579968285672225:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5164618986717855:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
double f(double a, double b, double c) {
        double r8128296 = b;
        double r8128297 = -r8128296;
        double r8128298 = r8128296 * r8128296;
        double r8128299 = 4.0;
        double r8128300 = a;
        double r8128301 = r8128299 * r8128300;
        double r8128302 = c;
        double r8128303 = r8128301 * r8128302;
        double r8128304 = r8128298 - r8128303;
        double r8128305 = sqrt(r8128304);
        double r8128306 = r8128297 + r8128305;
        double r8128307 = 2.0;
        double r8128308 = r8128307 * r8128300;
        double r8128309 = r8128306 / r8128308;
        return r8128309;
}
↓
double f(double a, double b, double c) {
        double r8128310 = b;
        double r8128311 = 0.003195690106913911;
        bool r8128312 = r8128310 <= r8128311;
        double r8128313 = r8128310 * r8128310;
        double r8128314 = 4.0;
        double r8128315 = c;
        double r8128316 = a;
        double r8128317 = r8128315 * r8128316;
        double r8128318 = r8128314 * r8128317;
        double r8128319 = r8128313 - r8128318;
        double r8128320 = sqrt(r8128319);
        double r8128321 = r8128319 * r8128320;
        double r8128322 = r8128313 * r8128310;
        double r8128323 = r8128321 - r8128322;
        double r8128324 = 2.0;
        double r8128325 = r8128324 * r8128316;
        double r8128326 = r8128310 * r8128320;
        double r8128327 = r8128313 + r8128326;
        double r8128328 = r8128320 * r8128320;
        double r8128329 = r8128327 + r8128328;
        double r8128330 = r8128325 * r8128329;
        double r8128331 = r8128323 / r8128330;
        double r8128332 = 0.1579968285672225;
        bool r8128333 = r8128310 <= r8128332;
        double r8128334 = r8128315 / r8128310;
        double r8128335 = -r8128334;
        double r8128336 = 1.5164618986717855;
        bool r8128337 = r8128310 <= r8128336;
        double r8128338 = r8128337 ? r8128331 : r8128335;
        double r8128339 = r8128333 ? r8128335 : r8128338;
        double r8128340 = r8128312 ? r8128331 : r8128339;
        return r8128340;
}


\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 0.003195690106913911:\\
\;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\

\mathbf{elif}\;b \le 0.1579968285672225:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.5164618986717855:\\
\;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\

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

\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.003195690106913911 or 0.1579968285672225 < b < 1.5164618986717855

    1. Initial program 23.5

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

    2. Simplified23.5

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

    4. Applied flip3--23.6

      \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{2 \cdot a}\]

    5. Applied associate-/l/23.6

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)\right)}}\]

    6. Simplified22.9

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

    if 0.003195690106913911 < b < 0.1579968285672225 or 1.5164618986717855 < b

    1. Initial program 46.5

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

    2. Simplified46.5

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

    3. Taylor expanded around inf 10.0

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

    4. Simplified10.0

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.003195690106913911:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{elif}\;b \le 0.1579968285672225:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.5164618986717855:\\ \;\;\;\;\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(2 \cdot a\right) \cdot \left(\left(b \cdot b + b \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))