Average Error: 34.1 → 10.6
Time: 13.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 46522626219735482368:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 46522626219735482368:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r76265 = b;
        double r76266 = -r76265;
        double r76267 = r76265 * r76265;
        double r76268 = 4.0;
        double r76269 = a;
        double r76270 = c;
        double r76271 = r76269 * r76270;
        double r76272 = r76268 * r76271;
        double r76273 = r76267 - r76272;
        double r76274 = sqrt(r76273);
        double r76275 = r76266 - r76274;
        double r76276 = 2.0;
        double r76277 = r76276 * r76269;
        double r76278 = r76275 / r76277;
        return r76278;
}
↓
double f(double a, double b, double c) {
        double r76279 = b;
        double r76280 = -6.151530580746178e-108;
        bool r76281 = r76279 <= r76280;
        double r76282 = -1.0;
        double r76283 = c;
        double r76284 = r76283 / r76279;
        double r76285 = r76282 * r76284;
        double r76286 = 4.652262621973548e+19;
        bool r76287 = r76279 <= r76286;
        double r76288 = 1.0;
        double r76289 = 2.0;
        double r76290 = a;
        double r76291 = r76289 * r76290;
        double r76292 = -r76279;
        double r76293 = r76279 * r76279;
        double r76294 = 4.0;
        double r76295 = r76290 * r76283;
        double r76296 = r76294 * r76295;
        double r76297 = r76293 - r76296;
        double r76298 = sqrt(r76297);
        double r76299 = r76292 - r76298;
        double r76300 = r76291 / r76299;
        double r76301 = r76288 / r76300;
        double r76302 = 1.0;
        double r76303 = r76279 / r76290;
        double r76304 = r76284 - r76303;
        double r76305 = r76302 * r76304;
        double r76306 = r76287 ? r76301 : r76305;
        double r76307 = r76281 ? r76285 : r76306;
        return r76307;
}

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

Original34.1
Target20.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.151530580746178e-108

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 10.7

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

    if -6.151530580746178e-108 < b < 4.652262621973548e+19

    1. Initial program 12.9

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

    3. Applied clear-num13.0

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

    if 4.652262621973548e+19 < b

    1. Initial program 34.5

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

    2. Taylor expanded around inf 6.9

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

    3. Simplified6.9

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.151530580746178328361254057251815139505 \cdot 10^{-108}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 46522626219735482368:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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)))