Average Error: 34.6 → 6.3
Time: 20.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\
\;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\

\mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\
\;\;\;\;\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 r80282 = b;
        double r80283 = -r80282;
        double r80284 = r80282 * r80282;
        double r80285 = 4.0;
        double r80286 = a;
        double r80287 = c;
        double r80288 = r80286 * r80287;
        double r80289 = r80285 * r80288;
        double r80290 = r80284 - r80289;
        double r80291 = sqrt(r80290);
        double r80292 = r80283 - r80291;
        double r80293 = 2.0;
        double r80294 = r80293 * r80286;
        double r80295 = r80292 / r80294;
        return r80295;
}
↓
double f(double a, double b, double c) {
        double r80296 = b;
        double r80297 = -5.263290697710818e+146;
        bool r80298 = r80296 <= r80297;
        double r80299 = -1.0;
        double r80300 = c;
        double r80301 = r80300 / r80296;
        double r80302 = r80299 * r80301;
        double r80303 = -2.182382645844659e-295;
        bool r80304 = r80296 <= r80303;
        double r80305 = 1.0;
        double r80306 = 2.0;
        double r80307 = a;
        double r80308 = r80306 * r80307;
        double r80309 = 4.0;
        double r80310 = r80309 * r80307;
        double r80311 = r80308 / r80310;
        double r80312 = r80296 * r80296;
        double r80313 = r80307 * r80300;
        double r80314 = r80309 * r80313;
        double r80315 = r80312 - r80314;
        double r80316 = sqrt(r80315);
        double r80317 = r80316 - r80296;
        double r80318 = r80300 / r80317;
        double r80319 = r80311 / r80318;
        double r80320 = r80305 / r80319;
        double r80321 = 3.1607591925776442e+143;
        bool r80322 = r80296 <= r80321;
        double r80323 = -r80296;
        double r80324 = r80323 - r80316;
        double r80325 = r80324 / r80308;
        double r80326 = 1.0;
        double r80327 = r80296 / r80307;
        double r80328 = r80301 - r80327;
        double r80329 = r80326 * r80328;
        double r80330 = r80322 ? r80325 : r80329;
        double r80331 = r80304 ? r80320 : r80330;
        double r80332 = r80298 ? r80302 : r80331;
        return r80332;
}

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

Original34.6
Target20.9
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.263290697710818e+146

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 1.3

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

    if -5.263290697710818e+146 < b < -2.182382645844659e-295

    1. Initial program 34.7

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

    3. Applied flip--34.7

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied clear-num15.9

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

    8. Simplified15.9

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

    10. Applied *-un-lft-identity15.9

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

    11. Applied times-frac13.5

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

    12. Applied associate-/r*7.6

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

    13. Simplified7.6

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

    if -2.182382645844659e-295 < b < 3.1607591925776442e+143

    1. Initial program 9.3

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

    if 3.1607591925776442e+143 < b

    1. Initial program 59.6

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))