Average Error: 33.7 → 8.8
Time: 17.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6733539759872003 \cdot 10^{120}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.6733539759872003 \cdot 10^{120}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r58249 = b;
        double r58250 = -r58249;
        double r58251 = r58249 * r58249;
        double r58252 = 4.0;
        double r58253 = a;
        double r58254 = c;
        double r58255 = r58253 * r58254;
        double r58256 = r58252 * r58255;
        double r58257 = r58251 - r58256;
        double r58258 = sqrt(r58257);
        double r58259 = r58250 + r58258;
        double r58260 = 2.0;
        double r58261 = r58260 * r58253;
        double r58262 = r58259 / r58261;
        return r58262;
}
↓
double f(double a, double b, double c) {
        double r58263 = b;
        double r58264 = -1.6733539759872003e+120;
        bool r58265 = r58263 <= r58264;
        double r58266 = 1.0;
        double r58267 = c;
        double r58268 = r58267 / r58263;
        double r58269 = a;
        double r58270 = r58263 / r58269;
        double r58271 = r58268 - r58270;
        double r58272 = r58266 * r58271;
        double r58273 = 2.3448135958717817e-114;
        bool r58274 = r58263 <= r58273;
        double r58275 = 1.0;
        double r58276 = 2.0;
        double r58277 = r58276 * r58269;
        double r58278 = r58263 * r58263;
        double r58279 = 4.0;
        double r58280 = r58269 * r58267;
        double r58281 = r58279 * r58280;
        double r58282 = r58278 - r58281;
        double r58283 = sqrt(r58282);
        double r58284 = r58283 - r58263;
        double r58285 = r58277 / r58284;
        double r58286 = r58275 / r58285;
        double r58287 = 3.3627913932282257e+22;
        bool r58288 = r58263 <= r58287;
        double r58289 = -r58263;
        double r58290 = sqrt(r58283);
        double r58291 = r58290 * r58290;
        double r58292 = r58289 - r58291;
        double r58293 = r58281 / r58292;
        double r58294 = r58293 / r58277;
        double r58295 = -1.0;
        double r58296 = r58295 * r58268;
        double r58297 = r58288 ? r58294 : r58296;
        double r58298 = r58274 ? r58286 : r58297;
        double r58299 = r58265 ? r58272 : r58298;
        return r58299;
}

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
Target20.8
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6733539759872003e+120

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.6733539759872003e+120 < b < 2.3448135958717817e-114

    1. Initial program 11.0

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

    3. Applied clear-num11.2

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

    4. Simplified11.2

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

    if 2.3448135958717817e-114 < b < 3.3627913932282257e+22

    1. Initial program 36.9

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

    3. Applied flip-+36.9

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

    4. Simplified16.0

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

    6. Applied add-sqr-sqrt16.0

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

    7. Applied sqrt-prod16.1

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

    if 3.3627913932282257e+22 < b

    1. Initial program 56.3

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

    2. Taylor expanded around inf 5.1

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6733539759872003 \cdot 10^{120}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.34481359587178172 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 3.36279139322822572 \cdot 10^{22}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))