Average Error: 34.4 → 7.2
Time: 5.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.1532693658084115 \cdot 10^{84}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.83946022919216105 \cdot 10^{-236}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \cdot c\right)\\ \mathbf{elif}\;b_2 \le 8.04930514182412967 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.1532693658084115 \cdot 10^{84}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.83946022919216105 \cdot 10^{-236}:\\
\;\;\;\;1 \cdot \left(\frac{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \cdot c\right)\\

\mathbf{elif}\;b_2 \le 8.04930514182412967 \cdot 10^{105}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r74937 = b_2;
        double r74938 = -r74937;
        double r74939 = r74937 * r74937;
        double r74940 = a;
        double r74941 = c;
        double r74942 = r74940 * r74941;
        double r74943 = r74939 - r74942;
        double r74944 = sqrt(r74943);
        double r74945 = r74938 - r74944;
        double r74946 = r74945 / r74940;
        return r74946;
}
↓
double f(double a, double b_2, double c) {
        double r74947 = b_2;
        double r74948 = -2.1532693658084115e+84;
        bool r74949 = r74947 <= r74948;
        double r74950 = -0.5;
        double r74951 = c;
        double r74952 = r74951 / r74947;
        double r74953 = r74950 * r74952;
        double r74954 = 1.839460229192161e-236;
        bool r74955 = r74947 <= r74954;
        double r74956 = 1.0;
        double r74957 = a;
        double r74958 = r74947 * r74947;
        double r74959 = r74957 * r74951;
        double r74960 = r74958 - r74959;
        double r74961 = sqrt(r74960);
        double r74962 = r74961 - r74947;
        double r74963 = r74957 / r74962;
        double r74964 = r74963 / r74957;
        double r74965 = r74964 * r74951;
        double r74966 = r74956 * r74965;
        double r74967 = 8.04930514182413e+105;
        bool r74968 = r74947 <= r74967;
        double r74969 = -r74947;
        double r74970 = r74969 - r74961;
        double r74971 = r74957 / r74970;
        double r74972 = r74956 / r74971;
        double r74973 = 0.5;
        double r74974 = r74973 * r74952;
        double r74975 = 2.0;
        double r74976 = r74947 / r74957;
        double r74977 = r74975 * r74976;
        double r74978 = r74974 - r74977;
        double r74979 = r74968 ? r74972 : r74978;
        double r74980 = r74955 ? r74966 : r74979;
        double r74981 = r74949 ? r74953 : r74980;
        return r74981;
}

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 < -2.1532693658084115e+84

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 2.9

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

    if -2.1532693658084115e+84 < b_2 < 1.839460229192161e-236

    1. Initial program 30.0

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

    3. Applied flip--30.1

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

    4. Simplified17.2

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

    5. Simplified17.2

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

    7. Applied *-un-lft-identity17.2

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

    8. Applied *-un-lft-identity17.2

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

    9. Applied times-frac17.2

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

    10. Simplified17.2

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

    11. Simplified14.7

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

    13. Applied div-inv14.7

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

    14. Applied associate-/r*15.1

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

    16. Applied *-un-lft-identity15.1

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

    17. Applied times-frac15.1

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

    18. Simplified15.1

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

    19. Simplified11.0

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

    if 1.839460229192161e-236 < b_2 < 8.04930514182413e+105

    1. Initial program 8.6

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

    3. Applied clear-num8.7

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

    if 8.04930514182413e+105 < b_2

    1. Initial program 49.7

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

    2. Taylor expanded around inf 3.8

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.1532693658084115 \cdot 10^{84}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.83946022919216105 \cdot 10^{-236}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \cdot c\right)\\ \mathbf{elif}\;b_2 \le 8.04930514182412967 \cdot 10^{105}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))