Average Error: 33.9 → 6.6
Time: 16.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{-2}\\ \mathbf{elif}\;b_2 \le \frac{2563685591414627}{3.630412374213337555659332270873780664383 \cdot 10^{280}}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\
\;\;\;\;\frac{\frac{c}{b_2}}{-2}\\

\mathbf{elif}\;b_2 \le \frac{2563685591414627}{3.630412374213337555659332270873780664383 \cdot 10^{280}}:\\
\;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r58678 = b_2;
        double r58679 = -r58678;
        double r58680 = r58678 * r58678;
        double r58681 = a;
        double r58682 = c;
        double r58683 = r58681 * r58682;
        double r58684 = r58680 - r58683;
        double r58685 = sqrt(r58684);
        double r58686 = r58679 - r58685;
        double r58687 = r58686 / r58681;
        return r58687;
}
↓
double f(double a, double b_2, double c) {
        double r58688 = b_2;
        double r58689 = -5.674469085146397e+110;
        bool r58690 = r58688 <= r58689;
        double r58691 = c;
        double r58692 = r58691 / r58688;
        double r58693 = -2.0;
        double r58694 = r58692 / r58693;
        double r58695 = 2563685591414627.0;
        double r58696 = 3.6304123742133376e+280;
        double r58697 = r58695 / r58696;
        bool r58698 = r58688 <= r58697;
        double r58699 = 1.0;
        double r58700 = r58688 * r58688;
        double r58701 = a;
        double r58702 = r58701 * r58691;
        double r58703 = r58700 - r58702;
        double r58704 = sqrt(r58703);
        double r58705 = r58704 - r58688;
        double r58706 = r58691 / r58705;
        double r58707 = r58699 * r58706;
        double r58708 = 1.7151811081882383e+78;
        bool r58709 = r58688 <= r58708;
        double r58710 = -r58688;
        double r58711 = r58710 - r58704;
        double r58712 = r58711 / r58701;
        double r58713 = r58688 / r58701;
        double r58714 = r58693 * r58713;
        double r58715 = r58709 ? r58712 : r58714;
        double r58716 = r58698 ? r58707 : r58715;
        double r58717 = r58690 ? r58694 : r58716;
        return r58717;
}

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 < -5.674469085146397e+110

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -5.674469085146397e+110 < b_2 < 7.061692521831336e-266

    1. Initial program 31.8

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

    3. Applied flip--31.8

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

    4. Simplified16.1

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

    5. Simplified16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied associate-/r*16.1

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

    9. Simplified14.1

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

    11. Applied *-un-lft-identity14.1

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

    12. Applied *-un-lft-identity14.1

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

    13. Applied times-frac14.1

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

    14. Applied associate-/l*14.0

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

    15. Simplified9.0

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

    17. Applied div-inv9.0

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

    18. Simplified8.7

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

    if 7.061692521831336e-266 < b_2 < 1.7151811081882383e+78

    1. Initial program 8.5

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

    if 1.7151811081882383e+78 < b_2

    1. Initial program 43.0

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

    3. Applied flip--62.6

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

    4. Simplified61.8

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

    5. Simplified61.8

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

    6. Taylor expanded around 0 4.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.674469085146396739103610609439188639717 \cdot 10^{110}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{-2}\\ \mathbf{elif}\;b_2 \le \frac{2563685591414627}{3.630412374213337555659332270873780664383 \cdot 10^{280}}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.715181108188238274259588142060201574853 \cdot 10^{78}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))