Average Error: 34.3 → 6.6
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\
\;\;\;\;\frac{2 \cdot c}{\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 r70127 = b;
        double r70128 = -r70127;
        double r70129 = r70127 * r70127;
        double r70130 = 4.0;
        double r70131 = a;
        double r70132 = c;
        double r70133 = r70131 * r70132;
        double r70134 = r70130 * r70133;
        double r70135 = r70129 - r70134;
        double r70136 = sqrt(r70135);
        double r70137 = r70128 + r70136;
        double r70138 = 2.0;
        double r70139 = r70138 * r70131;
        double r70140 = r70137 / r70139;
        return r70140;
}
↓
double f(double a, double b, double c) {
        double r70141 = b;
        double r70142 = -4.2975228517561493e+130;
        bool r70143 = r70141 <= r70142;
        double r70144 = 1.0;
        double r70145 = c;
        double r70146 = r70145 / r70141;
        double r70147 = a;
        double r70148 = r70141 / r70147;
        double r70149 = r70146 - r70148;
        double r70150 = r70144 * r70149;
        double r70151 = -1.9415295134599817e-260;
        bool r70152 = r70141 <= r70151;
        double r70153 = -r70141;
        double r70154 = r70141 * r70141;
        double r70155 = 4.0;
        double r70156 = r70147 * r70145;
        double r70157 = r70155 * r70156;
        double r70158 = r70154 - r70157;
        double r70159 = sqrt(r70158);
        double r70160 = r70153 + r70159;
        double r70161 = 1.0;
        double r70162 = 2.0;
        double r70163 = r70162 * r70147;
        double r70164 = r70161 / r70163;
        double r70165 = r70160 * r70164;
        double r70166 = 1.7521504885677008e+103;
        bool r70167 = r70141 <= r70166;
        double r70168 = r70162 * r70145;
        double r70169 = r70153 - r70159;
        double r70170 = r70168 / r70169;
        double r70171 = -1.0;
        double r70172 = r70171 * r70146;
        double r70173 = r70167 ? r70170 : r70172;
        double r70174 = r70152 ? r70165 : r70173;
        double r70175 = r70143 ? r70150 : r70174;
        return r70175;
}

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

Original34.3
Target20.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.2975228517561493e+130

    1. Initial program 54.6

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -4.2975228517561493e+130 < b < -1.9415295134599817e-260

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if -1.9415295134599817e-260 < b < 1.7521504885677008e+103

    1. Initial program 32.0

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

    3. Applied flip-+32.0

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

    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 div-inv16.4

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

    7. Applied associate-/l*21.0

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

    8. Simplified21.0

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

    10. Applied associate-/r*15.4

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

    11. Simplified15.3

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

    12. Taylor expanded around 0 9.0

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

    if 1.7521504885677008e+103 < b

    1. Initial program 59.7

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.297522851756149307625287590446857172218 \cdot 10^{130}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.941529513459981659526481896938835655376 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.752150488567700802544439975375947245276 \cdot 10^{103}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(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)))