Average Error: 33.2 → 9.7
Time: 53.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.232834657182634 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.958298658834631 \cdot 10^{-149}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le 3.8685970339297164 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.232834657182634 \cdot 10^{+85}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.958298658834631 \cdot 10^{-149}:\\
\;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\

\mathbf{elif}\;b_2 \le 3.8685970339297164 \cdot 10^{+90}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r3708646 = b_2;
        double r3708647 = -r3708646;
        double r3708648 = r3708646 * r3708646;
        double r3708649 = a;
        double r3708650 = c;
        double r3708651 = r3708649 * r3708650;
        double r3708652 = r3708648 - r3708651;
        double r3708653 = sqrt(r3708652);
        double r3708654 = r3708647 - r3708653;
        double r3708655 = r3708654 / r3708649;
        return r3708655;
}
↓
double f(double a, double b_2, double c) {
        double r3708656 = b_2;
        double r3708657 = -4.232834657182634e+85;
        bool r3708658 = r3708656 <= r3708657;
        double r3708659 = -0.5;
        double r3708660 = c;
        double r3708661 = r3708660 / r3708656;
        double r3708662 = r3708659 * r3708661;
        double r3708663 = -2.958298658834631e-149;
        bool r3708664 = r3708656 <= r3708663;
        double r3708665 = a;
        double r3708666 = r3708660 * r3708665;
        double r3708667 = -r3708656;
        double r3708668 = r3708656 * r3708656;
        double r3708669 = r3708668 - r3708666;
        double r3708670 = sqrt(r3708669);
        double r3708671 = r3708667 + r3708670;
        double r3708672 = r3708665 * r3708671;
        double r3708673 = r3708666 / r3708672;
        double r3708674 = 3.8685970339297164e+90;
        bool r3708675 = r3708656 <= r3708674;
        double r3708676 = r3708667 - r3708670;
        double r3708677 = r3708676 / r3708665;
        double r3708678 = -2.0;
        double r3708679 = r3708656 / r3708665;
        double r3708680 = r3708678 * r3708679;
        double r3708681 = r3708675 ? r3708677 : r3708680;
        double r3708682 = r3708664 ? r3708673 : r3708681;
        double r3708683 = r3708658 ? r3708662 : r3708682;
        return r3708683;
}

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 < -4.232834657182634e+85

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 2.5

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

    if -4.232834657182634e+85 < b_2 < -2.958298658834631e-149

    1. Initial program 38.7

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

    3. Applied flip--38.8

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

    4. Applied associate-/l/41.9

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

    5. Simplified18.9

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

    if -2.958298658834631e-149 < b_2 < 3.8685970339297164e+90

    1. Initial program 11.9

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

    3. Applied clear-num12.0

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

    5. Applied *-un-lft-identity12.0

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

    6. Applied *-un-lft-identity12.0

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

    7. Applied distribute-lft-out--12.0

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

    8. Applied *-un-lft-identity12.0

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

    9. Applied times-frac12.0

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

    10. Applied add-cube-cbrt12.0

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

    11. Applied times-frac12.0

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

    12. Simplified12.0

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

    13. Simplified11.9

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

    if 3.8685970339297164e+90 < b_2

    1. Initial program 42.2

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

    3. Applied clear-num42.3

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

    4. Taylor expanded around 0 3.9

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.232834657182634 \cdot 10^{+85}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.958298658834631 \cdot 10^{-149}:\\ \;\;\;\;\frac{c \cdot a}{a \cdot \left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}\\ \mathbf{elif}\;b_2 \le 3.8685970339297164 \cdot 10^{+90}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))