Average Error: 33.9 → 9.7
Time: 16.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.968339104808995575492318760699662265421 \cdot 10^{-12}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.111806971508121285438180273428571147006 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.830344977336381507706273409215818882174 \cdot 10^{-99}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.207713874276650441727053820537472427931 \cdot 10^{93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.968339104808995575492318760699662265421 \cdot 10^{-12}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -4.111806971508121285438180273428571147006 \cdot 10^{-72}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le -2.830344977336381507706273409215818882174 \cdot 10^{-99}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.207713874276650441727053820537472427931 \cdot 10^{93}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r62304 = b;
        double r62305 = -r62304;
        double r62306 = r62304 * r62304;
        double r62307 = 4.0;
        double r62308 = a;
        double r62309 = c;
        double r62310 = r62308 * r62309;
        double r62311 = r62307 * r62310;
        double r62312 = r62306 - r62311;
        double r62313 = sqrt(r62312);
        double r62314 = r62305 - r62313;
        double r62315 = 2.0;
        double r62316 = r62315 * r62308;
        double r62317 = r62314 / r62316;
        return r62317;
}
↓
double f(double a, double b, double c) {
        double r62318 = b;
        double r62319 = -2.9683391048089956e-12;
        bool r62320 = r62318 <= r62319;
        double r62321 = -1.0;
        double r62322 = c;
        double r62323 = r62322 / r62318;
        double r62324 = r62321 * r62323;
        double r62325 = -4.1118069715081213e-72;
        bool r62326 = r62318 <= r62325;
        double r62327 = 2.0;
        double r62328 = pow(r62318, r62327);
        double r62329 = r62328 - r62328;
        double r62330 = 4.0;
        double r62331 = a;
        double r62332 = r62331 * r62322;
        double r62333 = r62330 * r62332;
        double r62334 = r62329 + r62333;
        double r62335 = r62318 * r62318;
        double r62336 = r62335 - r62333;
        double r62337 = sqrt(r62336);
        double r62338 = r62337 - r62318;
        double r62339 = r62334 / r62338;
        double r62340 = 2.0;
        double r62341 = r62340 * r62331;
        double r62342 = r62339 / r62341;
        double r62343 = -2.8303449773363815e-99;
        bool r62344 = r62318 <= r62343;
        double r62345 = 1.2077138742766504e+93;
        bool r62346 = r62318 <= r62345;
        double r62347 = -r62318;
        double r62348 = r62347 - r62337;
        double r62349 = r62348 / r62341;
        double r62350 = 1.0;
        double r62351 = r62318 / r62331;
        double r62352 = r62323 - r62351;
        double r62353 = r62350 * r62352;
        double r62354 = r62346 ? r62349 : r62353;
        double r62355 = r62344 ? r62324 : r62354;
        double r62356 = r62326 ? r62342 : r62355;
        double r62357 = r62320 ? r62324 : r62356;
        return r62357;
}

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

Original33.9
Target21.0
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.9683391048089956e-12 or -4.1118069715081213e-72 < b < -2.8303449773363815e-99

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 7.9

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

    if -2.9683391048089956e-12 < b < -4.1118069715081213e-72

    1. Initial program 38.2

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

    3. Applied flip--38.2

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

    4. Simplified17.7

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

    5. Simplified17.7

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

    if -2.8303449773363815e-99 < b < 1.2077138742766504e+93

    1. Initial program 12.6

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

    if 1.2077138742766504e+93 < b

    1. Initial program 43.8

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

    2. Taylor expanded around inf 3.7

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

    3. Simplified3.7

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.968339104808995575492318760699662265421 \cdot 10^{-12}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -4.111806971508121285438180273428571147006 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -2.830344977336381507706273409215818882174 \cdot 10^{-99}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.207713874276650441727053820537472427931 \cdot 10^{93}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))