Average Error: 34.9 → 8.8
Time: 22.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.18698358163757920529281413894268206376 \cdot 10^{50}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.2648782224884627498150304098269944547 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}\right) \cdot \frac{\frac{a}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}}{\sqrt[3]{a}}\\ \mathbf{elif}\;b_2 \le 2.750597423682242910828949792302222725036 \cdot 10^{107}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.18698358163757920529281413894268206376 \cdot 10^{50}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.2648782224884627498150304098269944547 \cdot 10^{-158}:\\
\;\;\;\;\left(\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}\right) \cdot \frac{\frac{a}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}}{\sqrt[3]{a}}\\

\mathbf{elif}\;b_2 \le 2.750597423682242910828949792302222725036 \cdot 10^{107}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r93951 = b_2;
        double r93952 = -r93951;
        double r93953 = r93951 * r93951;
        double r93954 = a;
        double r93955 = c;
        double r93956 = r93954 * r93955;
        double r93957 = r93953 - r93956;
        double r93958 = sqrt(r93957);
        double r93959 = r93952 - r93958;
        double r93960 = r93959 / r93954;
        return r93960;
}
↓
double f(double a, double b_2, double c) {
        double r93961 = b_2;
        double r93962 = -2.186983581637579e+50;
        bool r93963 = r93961 <= r93962;
        double r93964 = -0.5;
        double r93965 = c;
        double r93966 = r93965 / r93961;
        double r93967 = r93964 * r93966;
        double r93968 = -3.2648782224884627e-158;
        bool r93969 = r93961 <= r93968;
        double r93970 = 1.0;
        double r93971 = -r93965;
        double r93972 = a;
        double r93973 = r93961 * r93961;
        double r93974 = fma(r93971, r93972, r93973);
        double r93975 = sqrt(r93974);
        double r93976 = r93975 - r93961;
        double r93977 = sqrt(r93976);
        double r93978 = r93970 / r93977;
        double r93979 = cbrt(r93972);
        double r93980 = r93979 * r93979;
        double r93981 = r93978 / r93980;
        double r93982 = sqrt(r93977);
        double r93983 = r93965 / r93982;
        double r93984 = r93981 * r93983;
        double r93985 = r93972 / r93982;
        double r93986 = r93985 / r93979;
        double r93987 = r93984 * r93986;
        double r93988 = 2.750597423682243e+107;
        bool r93989 = r93961 <= r93988;
        double r93990 = -r93961;
        double r93991 = r93972 * r93965;
        double r93992 = r93973 - r93991;
        double r93993 = sqrt(r93992);
        double r93994 = r93990 - r93993;
        double r93995 = r93970 / r93972;
        double r93996 = r93994 * r93995;
        double r93997 = 0.5;
        double r93998 = -2.0;
        double r93999 = r93961 / r93972;
        double r94000 = r93998 * r93999;
        double r94001 = fma(r93997, r93966, r94000);
        double r94002 = r93989 ? r93996 : r94001;
        double r94003 = r93969 ? r93987 : r94002;
        double r94004 = r93963 ? r93967 : r94003;
        return r94004;
}

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 < -2.186983581637579e+50

    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 4.0

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

    if -2.186983581637579e+50 < b_2 < -3.2648782224884627e-158

    1. Initial program 37.2

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

    3. Applied flip--37.3

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

    4. Simplified17.8

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

    5. Simplified17.8

      \[\leadsto \frac{\frac{0 + c \cdot a}{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt18.6

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

    8. Applied add-sqr-sqrt18.6

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

    9. Applied *-un-lft-identity18.6

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

    10. Applied times-frac18.6

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

    11. Applied times-frac18.4

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

    12. Simplified18.4

      \[\leadsto \frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\frac{\frac{c \cdot a}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{\sqrt[3]{a}}}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity18.4

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

    15. Applied cbrt-prod18.4

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

    16. Applied add-sqr-sqrt18.4

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

    17. Applied sqrt-prod18.4

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

    18. Applied times-frac16.0

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

    19. Applied times-frac12.7

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

    20. Applied associate-*r*16.3

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

    21. Simplified16.3

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

    if -3.2648782224884627e-158 < b_2 < 2.750597423682243e+107

    1. Initial program 11.2

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

    3. Applied div-inv11.3

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

    if 2.750597423682243e+107 < b_2

    1. Initial program 49.6

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

    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

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.18698358163757920529281413894268206376 \cdot 10^{50}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.2648782224884627498150304098269944547 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}\right) \cdot \frac{\frac{a}{\sqrt{\sqrt{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}}{\sqrt[3]{a}}\\ \mathbf{elif}\;b_2 \le 2.750597423682242910828949792302222725036 \cdot 10^{107}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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))