Average Error: 34.1 → 10.8
Time: 12.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.203999247190537828410646770952979837777 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.330769455174493332636713829620953568317 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{elif}\;b \le 3.705250296078930544323375298653357812472 \cdot 10^{-110}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.870709081743431705652419250795660469434 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3.203999247190537828410646770952979837777 \cdot 10^{59}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.330769455174493332636713829620953568317 \cdot 10^{-168}:\\
\;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\

\mathbf{elif}\;b \le 3.705250296078930544323375298653357812472 \cdot 10^{-110}:\\
\;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.870709081743431705652419250795660469434 \cdot 10^{-63}:\\
\;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r86274 = b;
        double r86275 = -r86274;
        double r86276 = r86274 * r86274;
        double r86277 = 4.0;
        double r86278 = a;
        double r86279 = c;
        double r86280 = r86278 * r86279;
        double r86281 = r86277 * r86280;
        double r86282 = r86276 - r86281;
        double r86283 = sqrt(r86282);
        double r86284 = r86275 - r86283;
        double r86285 = 2.0;
        double r86286 = r86285 * r86278;
        double r86287 = r86284 / r86286;
        return r86287;
}
↓
double f(double a, double b, double c) {
        double r86288 = b;
        double r86289 = -3.203999247190538e+59;
        bool r86290 = r86288 <= r86289;
        double r86291 = -1.0;
        double r86292 = c;
        double r86293 = r86292 / r86288;
        double r86294 = r86291 * r86293;
        double r86295 = 5.330769455174493e-168;
        bool r86296 = r86288 <= r86295;
        double r86297 = 4.0;
        double r86298 = r86297 * r86292;
        double r86299 = r86288 * r86288;
        double r86300 = a;
        double r86301 = r86300 * r86292;
        double r86302 = r86297 * r86301;
        double r86303 = r86299 - r86302;
        double r86304 = sqrt(r86303);
        double r86305 = r86304 - r86288;
        double r86306 = r86298 / r86305;
        double r86307 = 2.0;
        double r86308 = r86306 / r86307;
        double r86309 = 3.7052502960789305e-110;
        bool r86310 = r86288 <= r86309;
        double r86311 = -2.0;
        double r86312 = r86311 * r86288;
        double r86313 = r86307 * r86300;
        double r86314 = r86312 / r86313;
        double r86315 = 1.8707090817434317e-63;
        bool r86316 = r86288 <= r86315;
        double r86317 = 1.0;
        double r86318 = r86288 / r86300;
        double r86319 = r86293 - r86318;
        double r86320 = r86317 * r86319;
        double r86321 = r86316 ? r86308 : r86320;
        double r86322 = r86310 ? r86314 : r86321;
        double r86323 = r86296 ? r86308 : r86322;
        double r86324 = r86290 ? r86294 : r86323;
        return r86324;
}

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
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.203999247190538e+59

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 3.8

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

    if -3.203999247190538e+59 < b < 5.330769455174493e-168 or 3.7052502960789305e-110 < b < 1.8707090817434317e-63

    1. Initial program 25.0

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

    3. Applied flip--27.2

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

    4. Simplified17.6

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

    5. Simplified17.6

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

    7. Applied clear-num17.7

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

    8. Simplified17.7

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

    10. Applied *-un-lft-identity17.7

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

    11. Applied times-frac17.7

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

    12. Simplified17.7

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

    13. Simplified12.4

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

    15. Applied div-inv12.4

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

    16. Simplified12.1

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

    if 5.330769455174493e-168 < b < 3.7052502960789305e-110

    1. Initial program 6.3

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

    3. Applied flip--21.5

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

    4. Simplified21.7

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

    5. Simplified21.7

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

    6. Taylor expanded around 0 44.9

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]

    if 1.8707090817434317e-63 < b

    1. Initial program 28.0

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

    2. Taylor expanded around inf 11.0

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

    3. Simplified11.0

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.203999247190537828410646770952979837777 \cdot 10^{59}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.330769455174493332636713829620953568317 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{elif}\;b \le 3.705250296078930544323375298653357812472 \cdot 10^{-110}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.870709081743431705652419250795660469434 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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)))