Average Error: 34.2 → 9.1
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -763129212434271441067123993682640896:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\
\;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{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 r61358 = b;
        double r61359 = -r61358;
        double r61360 = r61358 * r61358;
        double r61361 = 4.0;
        double r61362 = a;
        double r61363 = c;
        double r61364 = r61362 * r61363;
        double r61365 = r61361 * r61364;
        double r61366 = r61360 - r61365;
        double r61367 = sqrt(r61366);
        double r61368 = r61359 - r61367;
        double r61369 = 2.0;
        double r61370 = r61369 * r61362;
        double r61371 = r61368 / r61370;
        return r61371;
}
↓
double f(double a, double b, double c) {
        double r61372 = b;
        double r61373 = -7.631292124342714e+35;
        bool r61374 = r61372 <= r61373;
        double r61375 = -1.0;
        double r61376 = c;
        double r61377 = r61376 / r61372;
        double r61378 = r61375 * r61377;
        double r61379 = 9.580019013081131e-278;
        bool r61380 = r61372 <= r61379;
        double r61381 = 4.0;
        double r61382 = a;
        double r61383 = r61381 * r61382;
        double r61384 = r61376 * r61383;
        double r61385 = 2.0;
        double r61386 = pow(r61372, r61385);
        double r61387 = r61382 * r61376;
        double r61388 = r61381 * r61387;
        double r61389 = r61386 - r61388;
        double r61390 = sqrt(r61389);
        double r61391 = r61390 - r61372;
        double r61392 = r61384 / r61391;
        double r61393 = 2.0;
        double r61394 = r61393 * r61382;
        double r61395 = r61392 / r61394;
        double r61396 = 5.031608061939103e+53;
        bool r61397 = r61372 <= r61396;
        double r61398 = -r61372;
        double r61399 = r61372 * r61372;
        double r61400 = r61399 - r61388;
        double r61401 = sqrt(r61400);
        double r61402 = r61398 - r61401;
        double r61403 = 1.0;
        double r61404 = r61403 / r61394;
        double r61405 = r61402 * r61404;
        double r61406 = 1.0;
        double r61407 = r61372 / r61382;
        double r61408 = r61377 - r61407;
        double r61409 = r61406 * r61408;
        double r61410 = r61397 ? r61405 : r61409;
        double r61411 = r61380 ? r61395 : r61410;
        double r61412 = r61374 ? r61378 : r61411;
        return r61412;
}

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

Original34.2
Target21.3
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.631292124342714e+35

    1. Initial program 56.2

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

    2. Taylor expanded around -inf 4.5

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

    if -7.631292124342714e+35 < b < 9.580019013081131e-278

    1. Initial program 27.7

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

    3. Applied flip--27.7

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

    4. Simplified16.7

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

    5. Simplified16.7

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

    if 9.580019013081131e-278 < b < 5.031608061939103e+53

    1. Initial program 9.4

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

    3. Applied div-inv9.6

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

    if 5.031608061939103e+53 < b

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

    3. Simplified5.7

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -763129212434271441067123993682640896:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.580019013081130749755184029236910886016 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))