Average Error: 34.9 → 8.8
Time: 12.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.546317250153943051696268494209933101223 \cdot 10^{99}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{1119122677593389}{1.372959532026121942996380159816278643454 \cdot 10^{157}}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{elif}\;b \le 19249548690054332:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.546317250153943051696268494209933101223 \cdot 10^{99}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le \frac{1119122677593389}{1.372959532026121942996380159816278643454 \cdot 10^{157}}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{elif}\;b \le 19249548690054332:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r72171 = b;
        double r72172 = -r72171;
        double r72173 = r72171 * r72171;
        double r72174 = 4.0;
        double r72175 = a;
        double r72176 = c;
        double r72177 = r72175 * r72176;
        double r72178 = r72174 * r72177;
        double r72179 = r72173 - r72178;
        double r72180 = sqrt(r72179);
        double r72181 = r72172 + r72180;
        double r72182 = 2.0;
        double r72183 = r72182 * r72175;
        double r72184 = r72181 / r72183;
        return r72184;
}
↓
double f(double a, double b, double c) {
        double r72185 = b;
        double r72186 = -3.546317250153943e+99;
        bool r72187 = r72185 <= r72186;
        double r72188 = 1.0;
        double r72189 = c;
        double r72190 = r72189 / r72185;
        double r72191 = a;
        double r72192 = r72185 / r72191;
        double r72193 = r72190 - r72192;
        double r72194 = r72188 * r72193;
        double r72195 = 1119122677593389.0;
        double r72196 = 1.372959532026122e+157;
        double r72197 = r72195 / r72196;
        bool r72198 = r72185 <= r72197;
        double r72199 = -r72185;
        double r72200 = r72185 * r72185;
        double r72201 = 4.0;
        double r72202 = r72191 * r72189;
        double r72203 = r72201 * r72202;
        double r72204 = r72200 - r72203;
        double r72205 = sqrt(r72204);
        double r72206 = r72199 + r72205;
        double r72207 = 2.0;
        double r72208 = r72206 / r72207;
        double r72209 = r72208 / r72191;
        double r72210 = 19249548690054332.0;
        bool r72211 = r72185 <= r72210;
        double r72212 = 1.0;
        double r72213 = r72207 * r72191;
        double r72214 = r72213 / r72203;
        double r72215 = r72199 - r72205;
        double r72216 = r72214 * r72215;
        double r72217 = r72212 / r72216;
        double r72218 = -1.0;
        double r72219 = r72218 * r72190;
        double r72220 = r72211 ? r72217 : r72219;
        double r72221 = r72198 ? r72209 : r72220;
        double r72222 = r72187 ? r72194 : r72221;
        return r72222;
}

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

Original34.9
Target21.7
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 < -3.546317250153943e+99

    1. Initial program 46.9

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

    2. Taylor expanded around -inf 3.7

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

    3. Simplified3.7

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

    if -3.546317250153943e+99 < b < 8.151170165531846e-143

    1. Initial program 11.9

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

    3. Applied associate-/r*11.9

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

    if 8.151170165531846e-143 < b < 19249548690054332.0

    1. Initial program 35.2

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

    3. Applied flip-+35.2

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

    4. Simplified16.3

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

    6. Applied *-un-lft-identity16.3

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

    7. Applied *-un-lft-identity16.3

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

    8. Applied times-frac16.3

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

    9. Applied associate-/l*16.4

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

    10. Simplified16.3

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

    if 19249548690054332.0 < b

    1. Initial program 56.9

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.546317250153943051696268494209933101223 \cdot 10^{99}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le \frac{1119122677593389}{1.372959532026121942996380159816278643454 \cdot 10^{157}}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{elif}\;b \le 19249548690054332:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))