Average Error: 34.1 → 6.9
Time: 11.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.886390593289588265884886582194784365324 \cdot 10^{-243}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.567076634438329703579154713319843964535 \cdot 10^{99}:\\ \;\;\;\;1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -3.886390593289588265884886582194784365324 \cdot 10^{-243}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 1.567076634438329703579154713319843964535 \cdot 10^{99}:\\
\;\;\;\;1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{\sqrt[3]{a}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r44090 = b_2;
        double r44091 = -r44090;
        double r44092 = r44090 * r44090;
        double r44093 = a;
        double r44094 = c;
        double r44095 = r44093 * r44094;
        double r44096 = r44092 - r44095;
        double r44097 = sqrt(r44096);
        double r44098 = r44091 + r44097;
        double r44099 = r44098 / r44093;
        return r44099;
}
↓
double f(double a, double b_2, double c) {
        double r44100 = b_2;
        double r44101 = -3.974595954042362e+78;
        bool r44102 = r44100 <= r44101;
        double r44103 = 0.5;
        double r44104 = c;
        double r44105 = r44104 / r44100;
        double r44106 = r44103 * r44105;
        double r44107 = 2.0;
        double r44108 = a;
        double r44109 = r44100 / r44108;
        double r44110 = r44107 * r44109;
        double r44111 = r44106 - r44110;
        double r44112 = -3.8863905932895883e-243;
        bool r44113 = r44100 <= r44112;
        double r44114 = -r44100;
        double r44115 = r44100 * r44100;
        double r44116 = r44108 * r44104;
        double r44117 = r44115 - r44116;
        double r44118 = sqrt(r44117);
        double r44119 = r44114 + r44118;
        double r44120 = 1.0;
        double r44121 = r44120 / r44108;
        double r44122 = r44119 * r44121;
        double r44123 = 1.5670766344383297e+99;
        bool r44124 = r44100 <= r44123;
        double r44125 = cbrt(r44108);
        double r44126 = r44114 - r44118;
        double r44127 = r44126 / r44104;
        double r44128 = r44125 / r44127;
        double r44129 = r44128 / r44125;
        double r44130 = r44120 * r44129;
        double r44131 = -0.5;
        double r44132 = r44131 * r44105;
        double r44133 = r44124 ? r44130 : r44132;
        double r44134 = r44113 ? r44122 : r44133;
        double r44135 = r44102 ? r44111 : r44134;
        return r44135;
}

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 < -3.974595954042362e+78

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 4.2

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

    if -3.974595954042362e+78 < b_2 < -3.8863905932895883e-243

    1. Initial program 7.9

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

    3. Applied div-inv8.0

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

    if -3.8863905932895883e-243 < b_2 < 1.5670766344383297e+99

    1. Initial program 31.1

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

    3. Applied flip-+31.1

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

    4. 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. Using strategy rm

    6. Applied *-un-lft-identity16.3

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

    7. Applied associate-/r*16.3

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

    8. Simplified14.2

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

    10. Applied add-cube-cbrt14.9

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

    11. Applied *-un-lft-identity14.9

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

    12. Applied *-un-lft-identity14.9

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

    13. Applied times-frac14.9

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

    14. Applied add-cube-cbrt14.2

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

    15. Applied times-frac14.3

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

    16. Applied times-frac10.5

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

    17. Simplified10.5

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

    if 1.5670766344383297e+99 < b_2

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.886390593289588265884886582194784365324 \cdot 10^{-243}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 1.567076634438329703579154713319843964535 \cdot 10^{99}:\\ \;\;\;\;1 \cdot \frac{\frac{\sqrt[3]{a}}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))