Average Error: 34.5 → 8.9
Time: 19.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.56119078388346221142618554723736351361 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c \cdot a}{\sqrt[3]{a} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} - b_2}}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -4.56119078388346221142618554723736351361 \cdot 10^{-123}:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c \cdot a}{\sqrt[3]{a} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} - b_2}}\\

\mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\
\;\;\;\;\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r88916 = b_2;
        double r88917 = -r88916;
        double r88918 = r88916 * r88916;
        double r88919 = a;
        double r88920 = c;
        double r88921 = r88919 * r88920;
        double r88922 = r88918 - r88921;
        double r88923 = sqrt(r88922);
        double r88924 = r88917 - r88923;
        double r88925 = r88924 / r88919;
        return r88925;
}
↓
double f(double a, double b_2, double c) {
        double r88926 = b_2;
        double r88927 = -3.1028950157805323e+69;
        bool r88928 = r88926 <= r88927;
        double r88929 = -0.5;
        double r88930 = c;
        double r88931 = r88930 / r88926;
        double r88932 = r88929 * r88931;
        double r88933 = -4.561190783883462e-123;
        bool r88934 = r88926 <= r88933;
        double r88935 = 1.0;
        double r88936 = 2.0;
        double r88937 = pow(r88926, r88936);
        double r88938 = a;
        double r88939 = r88938 * r88930;
        double r88940 = r88937 - r88939;
        double r88941 = sqrt(r88940);
        double r88942 = r88941 - r88926;
        double r88943 = cbrt(r88942);
        double r88944 = r88943 * r88943;
        double r88945 = r88935 / r88944;
        double r88946 = cbrt(r88938);
        double r88947 = r88946 * r88946;
        double r88948 = r88945 / r88947;
        double r88949 = r88930 * r88938;
        double r88950 = cbrt(r88941);
        double r88951 = r88950 * r88950;
        double r88952 = r88951 * r88950;
        double r88953 = r88952 - r88926;
        double r88954 = cbrt(r88953);
        double r88955 = r88946 * r88954;
        double r88956 = r88949 / r88955;
        double r88957 = r88948 * r88956;
        double r88958 = 2.1255630798514387e+135;
        bool r88959 = r88926 <= r88958;
        double r88960 = r88926 * r88926;
        double r88961 = r88960 - r88939;
        double r88962 = sqrt(r88961);
        double r88963 = r88926 + r88962;
        double r88964 = -r88938;
        double r88965 = r88963 / r88964;
        double r88966 = 0.5;
        double r88967 = r88926 / r88938;
        double r88968 = -2.0;
        double r88969 = r88967 * r88968;
        double r88970 = fma(r88966, r88931, r88969);
        double r88971 = r88959 ? r88965 : r88970;
        double r88972 = r88934 ? r88957 : r88971;
        double r88973 = r88928 ? r88932 : r88972;
        return r88973;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.1028950157805323e+69

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.1

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

    if -3.1028950157805323e+69 < b_2 < -4.561190783883462e-123

    1. Initial program 40.2

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

    3. Applied flip--40.3

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

    4. Simplified16.3

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified16.3

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt17.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

    8. Applied add-cube-cbrt17.3

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    9. Applied *-un-lft-identity17.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot c\right)}}{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    10. Applied times-frac17.3

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}} \cdot \frac{0 + a \cdot c}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

    11. Applied times-frac16.8

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{0 + a \cdot c}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a}}}\]

    12. Simplified16.6

      \[\leadsto \frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\frac{c \cdot a}{\sqrt[3]{a} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\]
    13. Using strategy rm

    14. Applied add-cube-cbrt16.6

      \[\leadsto \frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c \cdot a}{\sqrt[3]{a} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}}} - b_2}}\]

    if -4.561190783883462e-123 < b_2 < 2.1255630798514387e+135

    1. Initial program 11.3

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

    3. Applied frac-2neg11.3

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

    4. Simplified11.3

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

    if 2.1255630798514387e+135 < b_2

    1. Initial program 58.2

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

    2. Taylor expanded around inf 3.0

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

    3. Simplified3.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.102895015780532348136946077262401346805 \cdot 10^{69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -4.56119078388346221142618554723736351361 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c \cdot a}{\sqrt[3]{a} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}}\right) \cdot \sqrt[3]{\sqrt{{b_2}^{2} - a \cdot c}} - b_2}}\\ \mathbf{elif}\;b_2 \le 2.125563079851438727208684227808951636731 \cdot 10^{135}:\\ \;\;\;\;\frac{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))