Average Error: 34.0 → 10.6
Time: 4.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\
\;\;\;\;\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 r55777 = b_2;
        double r55778 = -r55777;
        double r55779 = r55777 * r55777;
        double r55780 = a;
        double r55781 = c;
        double r55782 = r55780 * r55781;
        double r55783 = r55779 - r55782;
        double r55784 = sqrt(r55783);
        double r55785 = r55778 - r55784;
        double r55786 = r55785 / r55780;
        return r55786;
}
↓
double f(double a, double b_2, double c) {
        double r55787 = b_2;
        double r55788 = -2.731633690849518e-121;
        bool r55789 = r55787 <= r55788;
        double r55790 = -0.5;
        double r55791 = c;
        double r55792 = r55791 / r55787;
        double r55793 = r55790 * r55792;
        double r55794 = 1.0273828621120979e+63;
        bool r55795 = r55787 <= r55794;
        double r55796 = 1.0;
        double r55797 = a;
        double r55798 = -r55787;
        double r55799 = r55787 * r55787;
        double r55800 = r55797 * r55791;
        double r55801 = r55799 - r55800;
        double r55802 = sqrt(r55801);
        double r55803 = r55798 - r55802;
        double r55804 = r55797 / r55803;
        double r55805 = r55796 / r55804;
        double r55806 = 0.5;
        double r55807 = r55806 * r55792;
        double r55808 = 2.0;
        double r55809 = r55787 / r55797;
        double r55810 = r55808 * r55809;
        double r55811 = r55807 - r55810;
        double r55812 = r55795 ? r55805 : r55811;
        double r55813 = r55789 ? r55793 : r55812;
        return r55813;
}

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 3 regimes

  2. if b_2 < -2.731633690849518e-121

    1. Initial program 51.0

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

    2. Taylor expanded around -inf 11.5

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

    if -2.731633690849518e-121 < b_2 < 1.0273828621120979e+63

    1. Initial program 12.1

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

    3. Applied clear-num12.2

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

    if 1.0273828621120979e+63 < b_2

    1. Initial program 39.8

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

    2. Taylor expanded around inf 5.4

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.731633690849517820308375807349583220341 \cdot 10^{-121}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.02738286211209785784187544728837722875 \cdot 10^{63}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))