Average Error: 34.8 → 11.1
Time: 5.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.53173257864094428 \cdot 10^{38}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.3119329296566595 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.54629873571897363 \cdot 10^{71}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.53173257864094428 \cdot 10^{38}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.3119329296566595 \cdot 10^{-127}:\\
\;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.54629873571897363 \cdot 10^{71}:\\
\;\;\;\;\frac{\frac{1}{2 \cdot a} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r63152 = b;
        double r63153 = -r63152;
        double r63154 = r63152 * r63152;
        double r63155 = 4.0;
        double r63156 = a;
        double r63157 = c;
        double r63158 = r63156 * r63157;
        double r63159 = r63155 * r63158;
        double r63160 = r63154 - r63159;
        double r63161 = sqrt(r63160);
        double r63162 = r63153 + r63161;
        double r63163 = 2.0;
        double r63164 = r63163 * r63156;
        double r63165 = r63162 / r63164;
        return r63165;
}
↓
double f(double a, double b, double c) {
        double r63166 = b;
        double r63167 = -5.531732578640944e+38;
        bool r63168 = r63166 <= r63167;
        double r63169 = 1.0;
        double r63170 = c;
        double r63171 = r63170 / r63166;
        double r63172 = a;
        double r63173 = r63166 / r63172;
        double r63174 = r63171 - r63173;
        double r63175 = r63169 * r63174;
        double r63176 = -1.3119329296566595e-127;
        bool r63177 = r63166 <= r63176;
        double r63178 = 4.0;
        double r63179 = 2.0;
        double r63180 = pow(r63166, r63179);
        double r63181 = r63180 - r63180;
        double r63182 = r63172 * r63170;
        double r63183 = r63178 * r63182;
        double r63184 = r63181 + r63183;
        double r63185 = r63184 / r63182;
        double r63186 = r63178 / r63185;
        double r63187 = -r63166;
        double r63188 = r63166 * r63166;
        double r63189 = r63188 - r63183;
        double r63190 = sqrt(r63189);
        double r63191 = r63187 + r63190;
        double r63192 = r63186 * r63191;
        double r63193 = 2.0;
        double r63194 = r63193 * r63172;
        double r63195 = r63192 / r63194;
        double r63196 = 2.5462987357189736e+71;
        bool r63197 = r63166 <= r63196;
        double r63198 = 1.0;
        double r63199 = r63198 / r63194;
        double r63200 = r63199 * r63184;
        double r63201 = r63187 - r63190;
        double r63202 = r63200 / r63201;
        double r63203 = -1.0;
        double r63204 = r63203 * r63171;
        double r63205 = r63197 ? r63202 : r63204;
        double r63206 = r63177 ? r63195 : r63205;
        double r63207 = r63168 ? r63175 : r63206;
        return r63207;
}

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

Original34.8
Target21.2
Herbie11.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.531732578640944e+38

    1. Initial program 36.8

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

    2. Taylor expanded around -inf 6.6

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

    3. Simplified6.6

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

    if -5.531732578640944e+38 < b < -1.3119329296566595e-127

    1. Initial program 6.5

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

    3. Applied flip-+38.9

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

    4. Simplified38.9

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

    6. Applied flip--38.9

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

    7. Applied associate-/r/38.9

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

    8. Simplified16.6

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

    if -1.3119329296566595e-127 < b < 2.5462987357189736e+71

    1. Initial program 27.1

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

    3. Applied div-inv27.1

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

    5. Applied flip-+28.1

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

    6. Applied associate-*l/28.1

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

    7. Simplified17.4

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

    if 2.5462987357189736e+71 < b

    1. Initial program 58.6

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

    2. Taylor expanded around inf 2.7

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.53173257864094428 \cdot 10^{38}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.3119329296566595 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.54629873571897363 \cdot 10^{71}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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)))