Average Error: 34.7 → 6.9
Time: 6.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.3202538172935113 \cdot 10^{68}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\
\;\;\;\;\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 3.67086091268017442 \cdot 10^{125}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r63117 = b;
        double r63118 = -r63117;
        double r63119 = r63117 * r63117;
        double r63120 = 4.0;
        double r63121 = a;
        double r63122 = c;
        double r63123 = r63121 * r63122;
        double r63124 = r63120 * r63123;
        double r63125 = r63119 - r63124;
        double r63126 = sqrt(r63125);
        double r63127 = r63118 + r63126;
        double r63128 = 2.0;
        double r63129 = r63128 * r63121;
        double r63130 = r63127 / r63129;
        return r63130;
}
↓
double f(double a, double b, double c) {
        double r63131 = b;
        double r63132 = -2.3202538172935113e+68;
        bool r63133 = r63131 <= r63132;
        double r63134 = 1.0;
        double r63135 = c;
        double r63136 = r63135 / r63131;
        double r63137 = a;
        double r63138 = r63131 / r63137;
        double r63139 = r63136 - r63138;
        double r63140 = r63134 * r63139;
        double r63141 = -8.908835082502404e-161;
        bool r63142 = r63131 <= r63141;
        double r63143 = -r63131;
        double r63144 = r63131 * r63131;
        double r63145 = 4.0;
        double r63146 = r63137 * r63135;
        double r63147 = r63145 * r63146;
        double r63148 = r63144 - r63147;
        double r63149 = sqrt(r63148);
        double r63150 = r63143 + r63149;
        double r63151 = 1.0;
        double r63152 = 2.0;
        double r63153 = r63152 * r63137;
        double r63154 = r63151 / r63153;
        double r63155 = r63150 * r63154;
        double r63156 = 3.6708609126801744e+125;
        bool r63157 = r63131 <= r63156;
        double r63158 = r63143 - r63149;
        double r63159 = r63158 / r63145;
        double r63160 = r63151 / r63159;
        double r63161 = r63160 / r63152;
        double r63162 = r63151 / r63135;
        double r63163 = r63151 / r63162;
        double r63164 = r63161 * r63163;
        double r63165 = -1.0;
        double r63166 = r63165 * r63136;
        double r63167 = r63157 ? r63164 : r63166;
        double r63168 = r63142 ? r63155 : r63167;
        double r63169 = r63133 ? r63140 : r63168;
        return r63169;
}

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

Original34.7
Target20.9
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3202538172935113e+68

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 5.1

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

    3. Simplified5.1

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

    if -2.3202538172935113e+68 < b < -8.908835082502404e-161

    1. Initial program 6.3

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

    3. Applied div-inv6.5

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

    if -8.908835082502404e-161 < b < 3.6708609126801744e+125

    1. Initial program 29.9

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

    3. Applied flip-+30.1

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

    4. Simplified16.7

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

    6. Applied clear-num16.9

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

    7. Simplified16.9

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

    9. Applied div-inv17.5

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

    10. Applied add-sqr-sqrt17.5

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

    11. Applied times-frac17.3

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

    12. Applied times-frac16.4

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

    13. Simplified16.4

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

    14. Simplified15.7

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

    16. Applied clear-num15.8

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

    17. Simplified10.6

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

    if 3.6708609126801744e+125 < b

    1. Initial program 61.6

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))