Average Error: 33.6 → 10.6
Time: 17.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\
\;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r2830707 = b_2;
        double r2830708 = -r2830707;
        double r2830709 = r2830707 * r2830707;
        double r2830710 = a;
        double r2830711 = c;
        double r2830712 = r2830710 * r2830711;
        double r2830713 = r2830709 - r2830712;
        double r2830714 = sqrt(r2830713);
        double r2830715 = r2830708 - r2830714;
        double r2830716 = r2830715 / r2830710;
        return r2830716;
}
↓
double f(double a, double b_2, double c) {
        double r2830717 = b_2;
        double r2830718 = -7.363255598823911e-15;
        bool r2830719 = r2830717 <= r2830718;
        double r2830720 = -0.5;
        double r2830721 = c;
        double r2830722 = r2830721 / r2830717;
        double r2830723 = r2830720 * r2830722;
        double r2830724 = -1.823572975982288e-27;
        bool r2830725 = r2830717 <= r2830724;
        double r2830726 = r2830717 * r2830717;
        double r2830727 = a;
        double r2830728 = r2830721 * r2830727;
        double r2830729 = r2830726 - r2830728;
        double r2830730 = r2830726 - r2830729;
        double r2830731 = r2830730 / r2830727;
        double r2830732 = -r2830717;
        double r2830733 = sqrt(r2830729);
        double r2830734 = r2830732 + r2830733;
        double r2830735 = r2830731 / r2830734;
        double r2830736 = -2.3344326820285623e-123;
        bool r2830737 = r2830717 <= r2830736;
        double r2830738 = 1.6691257204922504e+85;
        bool r2830739 = r2830717 <= r2830738;
        double r2830740 = r2830732 - r2830733;
        double r2830741 = r2830740 / r2830727;
        double r2830742 = r2830717 / r2830727;
        double r2830743 = -2.0;
        double r2830744 = 2.0;
        double r2830745 = r2830722 / r2830744;
        double r2830746 = fma(r2830742, r2830743, r2830745);
        double r2830747 = r2830739 ? r2830741 : r2830746;
        double r2830748 = r2830737 ? r2830723 : r2830747;
        double r2830749 = r2830725 ? r2830735 : r2830748;
        double r2830750 = r2830719 ? r2830723 : r2830749;
        return r2830750;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -7.363255598823911e-15 or -1.823572975982288e-27 < b_2 < -2.3344326820285623e-123

    1. Initial program 50.8

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

    2. Taylor expanded around -inf 10.6

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

    if -7.363255598823911e-15 < b_2 < -1.823572975982288e-27

    1. Initial program 36.1

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

    3. Applied clear-num36.1

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

    5. Applied flip--36.2

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

    6. Applied associate-/r/36.2

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

    7. Applied associate-/r*36.2

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

    8. Simplified36.1

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

    if -2.3344326820285623e-123 < b_2 < 1.6691257204922504e+85

    1. Initial program 12.6

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

    3. Applied div-inv12.7

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

    5. Applied un-div-inv12.6

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

    if 1.6691257204922504e+85 < b_2

    1. Initial program 42.9

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

    3. Applied div-inv42.9

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

    5. Applied un-div-inv42.9

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

    6. Taylor expanded around inf 3.6

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

    7. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.823572975982288 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 - c \cdot a\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))