Average Error: 34.1 → 6.7
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\

\mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r92184 = b;
        double r92185 = -r92184;
        double r92186 = r92184 * r92184;
        double r92187 = 4.0;
        double r92188 = a;
        double r92189 = r92187 * r92188;
        double r92190 = c;
        double r92191 = r92189 * r92190;
        double r92192 = r92186 - r92191;
        double r92193 = sqrt(r92192);
        double r92194 = r92185 + r92193;
        double r92195 = 2.0;
        double r92196 = r92195 * r92188;
        double r92197 = r92194 / r92196;
        return r92197;
}
↓
double f(double a, double b, double c) {
        double r92198 = b;
        double r92199 = -2.2237630570465103e+109;
        bool r92200 = r92198 <= r92199;
        double r92201 = 1.0;
        double r92202 = c;
        double r92203 = r92202 / r92198;
        double r92204 = a;
        double r92205 = r92198 / r92204;
        double r92206 = r92203 - r92205;
        double r92207 = r92201 * r92206;
        double r92208 = -3.3193805664383666e-186;
        bool r92209 = r92198 <= r92208;
        double r92210 = -r92198;
        double r92211 = r92198 * r92198;
        double r92212 = 4.0;
        double r92213 = r92212 * r92204;
        double r92214 = r92213 * r92202;
        double r92215 = r92211 - r92214;
        double r92216 = sqrt(r92215);
        double r92217 = r92210 + r92216;
        double r92218 = sqrt(r92217);
        double r92219 = 2.0;
        double r92220 = r92218 / r92219;
        double r92221 = r92218 / r92204;
        double r92222 = r92220 * r92221;
        double r92223 = 1.458057835821772e+144;
        bool r92224 = r92198 <= r92223;
        double r92225 = 1.0;
        double r92226 = r92219 / r92212;
        double r92227 = r92225 / r92226;
        double r92228 = r92227 * r92202;
        double r92229 = r92210 - r92216;
        double r92230 = r92228 / r92229;
        double r92231 = -1.0;
        double r92232 = r92231 * r92203;
        double r92233 = r92224 ? r92230 : r92232;
        double r92234 = r92209 ? r92222 : r92233;
        double r92235 = r92200 ? r92207 : r92234;
        return r92235;
}

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
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.2237630570465103e+109

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -2.2237630570465103e+109 < b < -3.3193805664383666e-186

    1. Initial program 6.9

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

    3. Applied add-sqr-sqrt7.3

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

    4. Applied times-frac7.3

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

    if -3.3193805664383666e-186 < b < 1.458057835821772e+144

    1. Initial program 31.3

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

    3. Applied flip-+31.5

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

    4. Simplified16.1

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

    6. Applied *-un-lft-identity16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac16.1

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

    9. Applied associate-/l*16.3

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

    10. Simplified15.3

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

    12. Applied times-frac15.3

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

    13. Simplified10.2

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

    15. Applied associate-/r*9.9

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

    16. Simplified9.8

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

    if 1.458057835821772e+144 < b

    1. Initial program 62.9

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

    2. Taylor expanded around inf 1.5

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))