Average Error: 34.4 → 9.0
Time: 13.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.581291473695029702690078079656779261192 \cdot 10^{58}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -4.749476137934256724197446467547353552613 \cdot 10^{-147}:\\ \;\;\;\;\frac{\left(c \cdot a + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le 6.382246919184671949353170697340125221139 \cdot 10^{97}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} + \frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.581291473695029702690078079656779261192 \cdot 10^{58}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b_2 \le -4.749476137934256724197446467547353552613 \cdot 10^{-147}:\\
\;\;\;\;\frac{\left(c \cdot a + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{elif}\;b_2 \le 6.382246919184671949353170697340125221139 \cdot 10^{97}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23044 = b_2;
        double r23045 = -r23044;
        double r23046 = r23044 * r23044;
        double r23047 = a;
        double r23048 = c;
        double r23049 = r23047 * r23048;
        double r23050 = r23046 - r23049;
        double r23051 = sqrt(r23050);
        double r23052 = r23045 - r23051;
        double r23053 = r23052 / r23047;
        return r23053;
}
↓
double f(double a, double b_2, double c) {
        double r23054 = b_2;
        double r23055 = -2.5812914736950297e+58;
        bool r23056 = r23054 <= r23055;
        double r23057 = c;
        double r23058 = r23057 / r23054;
        double r23059 = -0.5;
        double r23060 = r23058 * r23059;
        double r23061 = -4.749476137934257e-147;
        bool r23062 = r23054 <= r23061;
        double r23063 = a;
        double r23064 = r23057 * r23063;
        double r23065 = r23054 * r23054;
        double r23066 = r23065 - r23065;
        double r23067 = r23064 + r23066;
        double r23068 = 1.0;
        double r23069 = r23068 / r23063;
        double r23070 = r23067 * r23069;
        double r23071 = -r23054;
        double r23072 = r23065 - r23064;
        double r23073 = sqrt(r23072);
        double r23074 = r23071 + r23073;
        double r23075 = r23070 / r23074;
        double r23076 = 6.382246919184672e+97;
        bool r23077 = r23054 <= r23076;
        double r23078 = r23071 / r23063;
        double r23079 = r23073 / r23063;
        double r23080 = r23078 - r23079;
        double r23081 = 0.5;
        double r23082 = r23054 / r23057;
        double r23083 = r23081 / r23082;
        double r23084 = -2.0;
        double r23085 = r23084 * r23054;
        double r23086 = r23085 / r23063;
        double r23087 = r23083 + r23086;
        double r23088 = r23077 ? r23080 : r23087;
        double r23089 = r23062 ? r23075 : r23088;
        double r23090 = r23056 ? r23060 : r23089;
        return r23090;
}

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 4 regimes

  2. if b_2 < -2.5812914736950297e+58

    1. Initial program 57.5

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

    2. Taylor expanded around -inf 3.4

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

    3. Simplified3.4

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

    if -2.5812914736950297e+58 < b_2 < -4.749476137934257e-147

    1. Initial program 37.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied clear-num37.7

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

    5. Applied flip--37.7

      \[\leadsto \frac{1}{\frac{a}{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}}\]

    6. Applied associate-/r/37.8

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}\]

    7. Applied associate-/r*37.7

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{a}{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    8. Simplified16.6

      \[\leadsto \frac{\color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right) \cdot \frac{1}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if -4.749476137934257e-147 < b_2 < 6.382246919184672e+97

    1. Initial program 11.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub11.8

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

    if 6.382246919184672e+97 < b_2

    1. Initial program 47.4

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

    2. Taylor expanded around inf 3.7

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

    3. Simplified3.9

      \[\leadsto \color{blue}{\frac{b_2 \cdot -2}{a} + \frac{\frac{1}{2}}{\frac{b_2}{c}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.581291473695029702690078079656779261192 \cdot 10^{58}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -4.749476137934256724197446467547353552613 \cdot 10^{-147}:\\ \;\;\;\;\frac{\left(c \cdot a + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le 6.382246919184671949353170697340125221139 \cdot 10^{97}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} + \frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))