Average Error: 33.8 → 7.2
Time: 25.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\ \mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\

\mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3359102 = b_2;
        double r3359103 = -r3359102;
        double r3359104 = r3359102 * r3359102;
        double r3359105 = a;
        double r3359106 = c;
        double r3359107 = r3359105 * r3359106;
        double r3359108 = r3359104 - r3359107;
        double r3359109 = sqrt(r3359108);
        double r3359110 = r3359103 - r3359109;
        double r3359111 = r3359110 / r3359105;
        return r3359111;
}
↓
double f(double a, double b_2, double c) {
        double r3359112 = b_2;
        double r3359113 = -6.565090470125855e+141;
        bool r3359114 = r3359112 <= r3359113;
        double r3359115 = -0.5;
        double r3359116 = c;
        double r3359117 = r3359116 / r3359112;
        double r3359118 = r3359115 * r3359117;
        double r3359119 = -5.587449545143923e-253;
        bool r3359120 = r3359112 <= r3359119;
        double r3359121 = 1.0;
        double r3359122 = r3359112 * r3359112;
        double r3359123 = a;
        double r3359124 = r3359116 * r3359123;
        double r3359125 = r3359122 - r3359124;
        double r3359126 = sqrt(r3359125);
        double r3359127 = r3359126 - r3359112;
        double r3359128 = r3359121 / r3359127;
        double r3359129 = cbrt(r3359123);
        double r3359130 = r3359128 / r3359129;
        double r3359131 = r3359116 * r3359129;
        double r3359132 = r3359130 * r3359131;
        double r3359133 = 9.336288915836175e+83;
        bool r3359134 = r3359112 <= r3359133;
        double r3359135 = -r3359112;
        double r3359136 = r3359135 - r3359126;
        double r3359137 = r3359121 / r3359123;
        double r3359138 = r3359136 * r3359137;
        double r3359139 = 0.5;
        double r3359140 = r3359139 * r3359117;
        double r3359141 = r3359112 / r3359123;
        double r3359142 = 2.0;
        double r3359143 = r3359141 * r3359142;
        double r3359144 = r3359140 - r3359143;
        double r3359145 = r3359134 ? r3359138 : r3359144;
        double r3359146 = r3359120 ? r3359132 : r3359145;
        double r3359147 = r3359114 ? r3359118 : r3359146;
        return r3359147;
}

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 < -6.565090470125855e+141

    1. Initial program 61.3

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

    2. Taylor expanded around -inf 1.7

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

    if -6.565090470125855e+141 < b_2 < -5.587449545143923e-253

    1. Initial program 36.7

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

    3. Applied flip--36.8

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    7. Applied add-cube-cbrt17.6

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

    8. Applied div-inv17.7

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

    9. Applied times-frac16.2

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

    10. Simplified13.5

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

    11. Taylor expanded around inf 37.8

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

    12. Simplified9.6

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

    if -5.587449545143923e-253 < b_2 < 9.336288915836175e+83

    1. Initial program 9.6

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

    3. Applied div-inv9.7

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

    if 9.336288915836175e+83 < b_2

    1. Initial program 42.5

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.565090470125855 \cdot 10^{+141}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -5.587449545143923 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\sqrt[3]{a}} \cdot \left(c \cdot \sqrt[3]{a}\right)\\ \mathbf{elif}\;b_2 \le 9.336288915836175 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))