Average Error: 33.7 → 9.0
Time: 22.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6239046376.848015:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -6239046376.848015:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\
\;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\

\mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r2013272 = b;
        double r2013273 = -r2013272;
        double r2013274 = r2013272 * r2013272;
        double r2013275 = 4.0;
        double r2013276 = a;
        double r2013277 = c;
        double r2013278 = r2013276 * r2013277;
        double r2013279 = r2013275 * r2013278;
        double r2013280 = r2013274 - r2013279;
        double r2013281 = sqrt(r2013280);
        double r2013282 = r2013273 - r2013281;
        double r2013283 = 2.0;
        double r2013284 = r2013283 * r2013276;
        double r2013285 = r2013282 / r2013284;
        return r2013285;
}
↓
double f(double a, double b, double c) {
        double r2013286 = b;
        double r2013287 = -6239046376.848015;
        bool r2013288 = r2013286 <= r2013287;
        double r2013289 = c;
        double r2013290 = -r2013289;
        double r2013291 = r2013290 / r2013286;
        double r2013292 = -2.396838732613953e-306;
        bool r2013293 = r2013286 <= r2013292;
        double r2013294 = 0.5;
        double r2013295 = a;
        double r2013296 = r2013294 / r2013295;
        double r2013297 = 4.0;
        double r2013298 = r2013289 * r2013297;
        double r2013299 = r2013298 * r2013295;
        double r2013300 = r2013296 * r2013299;
        double r2013301 = -r2013286;
        double r2013302 = r2013286 * r2013286;
        double r2013303 = r2013289 * r2013295;
        double r2013304 = r2013297 * r2013303;
        double r2013305 = r2013302 - r2013304;
        double r2013306 = sqrt(r2013305);
        double r2013307 = r2013301 + r2013306;
        double r2013308 = r2013300 / r2013307;
        double r2013309 = 9.179168538250646e+63;
        bool r2013310 = r2013286 <= r2013309;
        double r2013311 = r2013301 - r2013306;
        double r2013312 = r2013296 * r2013311;
        double r2013313 = r2013301 / r2013295;
        double r2013314 = r2013310 ? r2013312 : r2013313;
        double r2013315 = r2013293 ? r2013308 : r2013314;
        double r2013316 = r2013288 ? r2013291 : r2013315;
        return r2013316;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.7
Target20.7
Herbie9.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -6239046376.848015

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 5.1

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

    3. Simplified5.1

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

    if -6239046376.848015 < b < -2.396838732613953e-306

    1. Initial program 27.2

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

    3. Applied clear-num27.2

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

    5. Applied flip--27.3

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

    6. Applied associate-/r/27.3

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

    7. Applied associate-/r*27.3

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

    8. Simplified17.2

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

    if -2.396838732613953e-306 < b < 9.179168538250646e+63

    1. Initial program 9.1

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

    3. Applied clear-num9.3

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

    5. Applied associate-/r/9.3

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

    6. Simplified9.3

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

    if 9.179168538250646e+63 < b

    1. Initial program 37.8

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

    3. Applied clear-num37.9

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

    4. Taylor expanded around 0 6.1

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

    5. Simplified6.1

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6239046376.848015:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.396838732613953 \cdot 10^{-306}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot 4\right) \cdot a\right)}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \le 9.179168538250646 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))