Average Error: 1.7 → 0.5
Time: 17.6s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.14251708984375:\\ \;\;\;\;\frac{\frac{\left(b_2 + \left(-b_2\right)\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right) + c \cdot a}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -0.14251708984375:\\
\;\;\;\;\frac{\frac{\left(b_2 + \left(-b_2\right)\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right) + c \cdot a}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1128949 = b_2;
        double r1128950 = -r1128949;
        double r1128951 = r1128949 * r1128949;
        double r1128952 = a;
        double r1128953 = c;
        double r1128954 = r1128952 * r1128953;
        double r1128955 = r1128951 - r1128954;
        double r1128956 = sqrt(r1128955);
        double r1128957 = r1128950 - r1128956;
        double r1128958 = r1128957 / r1128952;
        return r1128958;
}
↓
double f(double a, double b_2, double c) {
        double r1128959 = b_2;
        double r1128960 = -0.14251708984375;
        bool r1128961 = r1128959 <= r1128960;
        double r1128962 = -r1128959;
        double r1128963 = r1128959 + r1128962;
        double r1128964 = r1128962 + r1128962;
        double r1128965 = r1128963 * r1128964;
        double r1128966 = c;
        double r1128967 = a;
        double r1128968 = r1128966 * r1128967;
        double r1128969 = r1128965 + r1128968;
        double r1128970 = r1128959 * r1128959;
        double r1128971 = r1128967 * r1128966;
        double r1128972 = r1128970 - r1128971;
        double r1128973 = sqrt(r1128972);
        double r1128974 = r1128962 + r1128973;
        double r1128975 = r1128969 / r1128974;
        double r1128976 = r1128975 / r1128967;
        double r1128977 = r1128962 - r1128973;
        double r1128978 = r1128977 / r1128967;
        double r1128979 = r1128961 ? r1128976 : r1128978;
        return r1128979;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b_2 < -0.14251708984375

    1. Initial program 3.6

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

    3. Applied p16-flip--3.2

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

    5. Applied difference-of-squares3.7

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

    7. Applied associate-/l*3.9

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

    9. Applied p16-flip--4.1

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

    10. Applied associate-/r/4.1

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

    11. Applied associate-/r*4.1

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

    12. Simplified0.7

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

    if -0.14251708984375 < b_2

    1. Initial program 0.4

      \[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.14251708984375:\\ \;\;\;\;\frac{\frac{\left(b_2 + \left(-b_2\right)\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right) + c \cdot a}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))