Average Error: 34.6 → 9.4
Time: 18.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\
\;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r66022 = b_2;
        double r66023 = -r66022;
        double r66024 = r66022 * r66022;
        double r66025 = a;
        double r66026 = c;
        double r66027 = r66025 * r66026;
        double r66028 = r66024 - r66027;
        double r66029 = sqrt(r66028);
        double r66030 = r66023 - r66029;
        double r66031 = r66030 / r66025;
        return r66031;
}
↓
double f(double a, double b_2, double c) {
        double r66032 = b_2;
        double r66033 = -1.2929563087095085e+102;
        bool r66034 = r66032 <= r66033;
        double r66035 = -0.5;
        double r66036 = c;
        double r66037 = r66036 / r66032;
        double r66038 = r66035 * r66037;
        double r66039 = -1.8936269564254704e-242;
        bool r66040 = r66032 <= r66039;
        double r66041 = 1.0;
        double r66042 = r66032 * r66032;
        double r66043 = a;
        double r66044 = r66043 * r66036;
        double r66045 = r66042 - r66044;
        double r66046 = sqrt(r66045);
        double r66047 = r66046 - r66032;
        double r66048 = r66047 / r66043;
        double r66049 = r66048 / r66036;
        double r66050 = r66041 / r66049;
        double r66051 = r66041 / r66043;
        double r66052 = r66050 * r66051;
        double r66053 = 0.010648423176581222;
        bool r66054 = r66032 <= r66053;
        double r66055 = -r66032;
        double r66056 = r66055 - r66046;
        double r66057 = r66056 / r66043;
        double r66058 = 0.5;
        double r66059 = r66058 * r66037;
        double r66060 = 2.0;
        double r66061 = r66032 / r66043;
        double r66062 = r66060 * r66061;
        double r66063 = r66059 - r66062;
        double r66064 = r66054 ? r66057 : r66063;
        double r66065 = r66040 ? r66052 : r66064;
        double r66066 = r66034 ? r66038 : r66065;
        return r66066;
}

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 < -1.2929563087095085e+102

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 2.5

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

    if -1.2929563087095085e+102 < b_2 < -1.8936269564254704e-242

    1. Initial program 34.8

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

    3. Applied flip--34.9

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied div-inv16.6

      \[\leadsto \color{blue}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2} \cdot \frac{1}{a}}\]
    8. Using strategy rm

    9. Applied clear-num16.9

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

    10. Simplified16.9

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

    12. Applied associate-/r*15.1

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

    if -1.8936269564254704e-242 < b_2 < 0.010648423176581222

    1. Initial program 11.4

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

    if 0.010648423176581222 < b_2

    1. Initial program 32.9

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

    2. Taylor expanded around inf 8.0

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.292956308709508535356330843875536768906 \cdot 10^{102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.893626956425470392922952841128924617441 \cdot 10^{-242}:\\ \;\;\;\;\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}} \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.01064842317658122247681085070780682144687:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))