Average Error: 34.0 → 9.3
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -14858297.0087451544:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\
\;\;\;\;\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}}{a}\\

\mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r78224 = b;
        double r78225 = -r78224;
        double r78226 = r78224 * r78224;
        double r78227 = 4.0;
        double r78228 = a;
        double r78229 = c;
        double r78230 = r78228 * r78229;
        double r78231 = r78227 * r78230;
        double r78232 = r78226 - r78231;
        double r78233 = sqrt(r78232);
        double r78234 = r78225 - r78233;
        double r78235 = 2.0;
        double r78236 = r78235 * r78228;
        double r78237 = r78234 / r78236;
        return r78237;
}
↓
double f(double a, double b, double c) {
        double r78238 = b;
        double r78239 = -14858297.008745154;
        bool r78240 = r78238 <= r78239;
        double r78241 = -1.0;
        double r78242 = c;
        double r78243 = r78242 / r78238;
        double r78244 = r78241 * r78243;
        double r78245 = -1.461539579389551e-137;
        bool r78246 = r78238 <= r78245;
        double r78247 = 2.0;
        double r78248 = pow(r78238, r78247);
        double r78249 = r78248 - r78248;
        double r78250 = 4.0;
        double r78251 = a;
        double r78252 = r78251 * r78242;
        double r78253 = r78250 * r78252;
        double r78254 = r78249 + r78253;
        double r78255 = r78238 * r78238;
        double r78256 = r78255 - r78253;
        double r78257 = sqrt(r78256);
        double r78258 = r78257 - r78238;
        double r78259 = r78254 / r78258;
        double r78260 = 2.0;
        double r78261 = r78259 / r78260;
        double r78262 = r78261 / r78251;
        double r78263 = 1.3845340503596435e+70;
        bool r78264 = r78238 <= r78263;
        double r78265 = -r78238;
        double r78266 = r78265 - r78257;
        double r78267 = r78266 / r78260;
        double r78268 = r78267 / r78251;
        double r78269 = r78238 / r78251;
        double r78270 = r78241 * r78269;
        double r78271 = r78264 ? r78268 : r78270;
        double r78272 = r78246 ? r78262 : r78271;
        double r78273 = r78240 ? r78244 : r78272;
        return r78273;
}

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

Original34.0
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -14858297.008745154

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 5.8

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

    if -14858297.008745154 < b < -1.461539579389551e-137

    1. Initial program 35.4

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

    3. Applied associate-/r*35.4

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

    5. Applied flip--35.5

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

    6. Simplified18.0

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

    7. Simplified18.0

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

    if -1.461539579389551e-137 < b < 1.3845340503596435e+70

    1. Initial program 11.6

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

    3. Applied associate-/r*11.6

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

    if 1.3845340503596435e+70 < b

    1. Initial program 41.5

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

    3. Applied associate-/r*41.5

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

    5. Applied *-un-lft-identity41.5

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

    6. Applied *-un-lft-identity41.5

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

    7. Applied times-frac41.5

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

    8. Applied associate-/l*41.6

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

    9. Taylor expanded around 0 5.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -14858297.0087451544:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.46153957938955092 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 1.3845340503596435 \cdot 10^{70}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))