Average Error: 33.5 → 10.4
Time: 18.9s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.037409549902714 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{\frac{b_2}{a}}{\frac{1}{2}}\\ \mathbf{elif}\;b_2 \le 1.9573510501313932 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.037409549902714 \cdot 10^{+148}:\\
\;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{\frac{b_2}{a}}{\frac{1}{2}}\\

\mathbf{elif}\;b_2 \le 1.9573510501313932 \cdot 10^{-67}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r863066 = b_2;
        double r863067 = -r863066;
        double r863068 = r863066 * r863066;
        double r863069 = a;
        double r863070 = c;
        double r863071 = r863069 * r863070;
        double r863072 = r863068 - r863071;
        double r863073 = sqrt(r863072);
        double r863074 = r863067 + r863073;
        double r863075 = r863074 / r863069;
        return r863075;
}
↓
double f(double a, double b_2, double c) {
        double r863076 = b_2;
        double r863077 = -6.037409549902714e+148;
        bool r863078 = r863076 <= r863077;
        double r863079 = 0.5;
        double r863080 = c;
        double r863081 = r863076 / r863080;
        double r863082 = r863079 / r863081;
        double r863083 = a;
        double r863084 = r863076 / r863083;
        double r863085 = r863084 / r863079;
        double r863086 = r863082 - r863085;
        double r863087 = 1.9573510501313932e-67;
        bool r863088 = r863076 <= r863087;
        double r863089 = r863076 * r863076;
        double r863090 = r863083 * r863080;
        double r863091 = r863089 - r863090;
        double r863092 = sqrt(r863091);
        double r863093 = r863092 - r863076;
        double r863094 = r863093 / r863083;
        double r863095 = -0.5;
        double r863096 = r863080 / r863076;
        double r863097 = r863095 * r863096;
        double r863098 = r863088 ? r863094 : r863097;
        double r863099 = r863078 ? r863086 : r863098;
        return r863099;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -6.037409549902714e+148

    1. Initial program 59.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified59.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv59.0

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied un-div-inv59.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    7. Using strategy rm

    8. Applied div-sub59.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]

    9. Taylor expanded around -inf 1.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    10. Simplified1.9

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{\frac{b_2}{a}}{\frac{1}{2}}}\]

    if -6.037409549902714e+148 < b_2 < 1.9573510501313932e-67

    1. Initial program 13.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified13.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv13.3

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]
    5. Using strategy rm

    6. Applied un-div-inv13.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    if 1.9573510501313932e-67 < b_2

    1. Initial program 51.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified51.9

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 9.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.037409549902714 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{\frac{b_2}{a}}{\frac{1}{2}}\\ \mathbf{elif}\;b_2 \le 1.9573510501313932 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))