Average Error: 34.6 → 6.3
Time: 19.2s
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 r64241 = b;
        double r64242 = -r64241;
        double r64243 = r64241 * r64241;
        double r64244 = 4.0;
        double r64245 = a;
        double r64246 = c;
        double r64247 = r64245 * r64246;
        double r64248 = r64244 * r64247;
        double r64249 = r64243 - r64248;
        double r64250 = sqrt(r64249);
        double r64251 = r64242 - r64250;
        double r64252 = 2.0;
        double r64253 = r64252 * r64245;
        double r64254 = r64251 / r64253;
        return r64254;
}
↓
double f(double a, double b, double c) {
        double r64255 = b;
        double r64256 = -5.263290697710818e+146;
        bool r64257 = r64255 <= r64256;
        double r64258 = -1.0;
        double r64259 = c;
        double r64260 = r64259 / r64255;
        double r64261 = r64258 * r64260;
        double r64262 = -2.182382645844659e-295;
        bool r64263 = r64255 <= r64262;
        double r64264 = 1.0;
        double r64265 = 2.0;
        double r64266 = a;
        double r64267 = r64265 * r64266;
        double r64268 = 4.0;
        double r64269 = r64268 * r64266;
        double r64270 = r64267 / r64269;
        double r64271 = r64255 * r64255;
        double r64272 = r64266 * r64259;
        double r64273 = r64268 * r64272;
        double r64274 = r64271 - r64273;
        double r64275 = sqrt(r64274);
        double r64276 = r64275 - r64255;
        double r64277 = r64259 / r64276;
        double r64278 = r64270 / r64277;
        double r64279 = r64264 / r64278;
        double r64280 = 3.1607591925776442e+143;
        bool r64281 = r64255 <= r64280;
        double r64282 = -r64255;
        double r64283 = r64282 - r64275;
        double r64284 = r64283 / r64267;
        double r64285 = 1.0;
        double r64286 = r64255 / r64266;
        double r64287 = r64260 - r64286;
        double r64288 = r64285 * r64287;
        double r64289 = r64281 ? r64284 : r64288;
        double r64290 = r64263 ? r64279 : r64289;
        double r64291 = r64257 ? r64261 : r64290;
        return r64291;
}

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 *-un-lft-identity15.7

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

    8. Applied *-un-lft-identity15.7

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

    9. Applied times-frac15.7

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

    10. Applied associate-/l*15.9

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

    11. Simplified15.9

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

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

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

    14. Applied times-frac13.5

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

    15. Applied associate-/r*7.6

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

    16. Simplified7.6

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

    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)))