Average Error: 33.8 → 9.3
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6472597296593428 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.0105231099196228 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 14169621.248013001:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.6472597296593428 \cdot 10^{81}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.0105231099196228 \cdot 10^{-270}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 14169621.248013001:\\
\;\;\;\;\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\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 r198131 = b;
        double r198132 = -r198131;
        double r198133 = r198131 * r198131;
        double r198134 = 4.0;
        double r198135 = a;
        double r198136 = r198134 * r198135;
        double r198137 = c;
        double r198138 = r198136 * r198137;
        double r198139 = r198133 - r198138;
        double r198140 = sqrt(r198139);
        double r198141 = r198132 + r198140;
        double r198142 = 2.0;
        double r198143 = r198142 * r198135;
        double r198144 = r198141 / r198143;
        return r198144;
}
↓
double f(double a, double b, double c) {
        double r198145 = b;
        double r198146 = -2.647259729659343e+81;
        bool r198147 = r198145 <= r198146;
        double r198148 = 1.0;
        double r198149 = c;
        double r198150 = r198149 / r198145;
        double r198151 = a;
        double r198152 = r198145 / r198151;
        double r198153 = r198150 - r198152;
        double r198154 = r198148 * r198153;
        double r198155 = 1.0105231099196228e-270;
        bool r198156 = r198145 <= r198155;
        double r198157 = 1.0;
        double r198158 = 2.0;
        double r198159 = r198158 * r198151;
        double r198160 = -r198145;
        double r198161 = r198145 * r198145;
        double r198162 = 4.0;
        double r198163 = r198162 * r198151;
        double r198164 = r198163 * r198149;
        double r198165 = r198161 - r198164;
        double r198166 = sqrt(r198165);
        double r198167 = r198160 + r198166;
        double r198168 = r198159 / r198167;
        double r198169 = r198157 / r198168;
        double r198170 = 14169621.248013001;
        bool r198171 = r198145 <= r198170;
        double r198172 = r198151 * r198149;
        double r198173 = r198162 * r198172;
        double r198174 = r198173 / r198158;
        double r198175 = r198174 / r198151;
        double r198176 = r198160 - r198166;
        double r198177 = r198175 / r198176;
        double r198178 = -1.0;
        double r198179 = r198178 * r198150;
        double r198180 = r198171 ? r198177 : r198179;
        double r198181 = r198156 ? r198169 : r198180;
        double r198182 = r198147 ? r198154 : r198181;
        return r198182;
}

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

Original33.8
Target21.2
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.647259729659343e+81

    1. Initial program 42.0

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

    2. Taylor expanded around -inf 4.7

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

    3. Simplified4.7

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

    if -2.647259729659343e+81 < b < 1.0105231099196228e-270

    1. Initial program 10.1

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

    3. Applied clear-num10.2

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

    if 1.0105231099196228e-270 < b < 14169621.248013001

    1. Initial program 27.3

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

    3. Applied flip-+27.3

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

    4. Simplified17.4

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

    6. Applied div-inv17.5

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

    7. Applied associate-/l*23.3

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

    8. Simplified23.2

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

    10. Applied associate-/r*17.4

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

    11. Simplified17.4

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

    if 14169621.248013001 < b

    1. Initial program 55.9

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

    2. Taylor expanded around inf 6.0

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6472597296593428 \cdot 10^{81}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.0105231099196228 \cdot 10^{-270}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 14169621.248013001:\\ \;\;\;\;\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(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)))