Average Error: 1.6 → 0.3
Time: 20.1s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le 0.0148773193359375:\\ \;\;\;\;\frac{1.0 \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]
\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 0.0148773193359375:\\
\;\;\;\;\frac{1.0 \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\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 r1224922 = b_2;
        double r1224923 = -r1224922;
        double r1224924 = r1224922 * r1224922;
        double r1224925 = a;
        double r1224926 = c;
        double r1224927 = r1224925 * r1224926;
        double r1224928 = r1224924 - r1224927;
        double r1224929 = sqrt(r1224928);
        double r1224930 = r1224923 - r1224929;
        double r1224931 = r1224930 / r1224925;
        return r1224931;
}
↓
double f(double a, double b_2, double c) {
        double r1224932 = b_2;
        double r1224933 = 0.0148773193359375;
        bool r1224934 = r1224932 <= r1224933;
        double r1224935 = 1.0;
        double r1224936 = c;
        double r1224937 = r1224935 * r1224936;
        double r1224938 = r1224932 * r1224932;
        double r1224939 = a;
        double r1224940 = r1224936 * r1224939;
        double r1224941 = r1224938 - r1224940;
        double r1224942 = sqrt(r1224941);
        double r1224943 = r1224942 - r1224932;
        double r1224944 = r1224937 / r1224943;
        double r1224945 = -r1224932;
        double r1224946 = r1224939 * r1224936;
        double r1224947 = r1224938 - r1224946;
        double r1224948 = sqrt(r1224947);
        double r1224949 = r1224945 - r1224948;
        double r1224950 = r1224949 / r1224939;
        double r1224951 = r1224934 ? r1224944 : r1224950;
        return r1224951;
}

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 < 0.0148773193359375

    1. Initial program 2.8

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

    3. Applied p16-flip--2.4

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

    4. Simplified0.6

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

    5. Simplified0.6

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(c \cdot a\right)}{\left(0.0\right)}\right)}{\left(a \cdot \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)\right)}}\]
    6. Using strategy rm

    7. Applied associate-/r*0.5

      \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(c \cdot a\right)}{\left(0.0\right)}\right)}{a}\right)}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}}\]

    8. Simplified0.3

      \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot c\right)}}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}\]

    if 0.0148773193359375 < b_2

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le 0.0148773193359375:\\ \;\;\;\;\frac{1.0 \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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))