Average Error: 33.1 → 8.0
Time: 1.1m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\
\;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r7667590 = b_2;
        double r7667591 = -r7667590;
        double r7667592 = r7667590 * r7667590;
        double r7667593 = a;
        double r7667594 = c;
        double r7667595 = r7667593 * r7667594;
        double r7667596 = r7667592 - r7667595;
        double r7667597 = sqrt(r7667596);
        double r7667598 = r7667591 - r7667597;
        double r7667599 = r7667598 / r7667593;
        return r7667599;
}
↓
double f(double a, double b_2, double c) {
        double r7667600 = b_2;
        double r7667601 = -6.473972066548491e+100;
        bool r7667602 = r7667600 <= r7667601;
        double r7667603 = -0.5;
        double r7667604 = c;
        double r7667605 = r7667604 / r7667600;
        double r7667606 = r7667603 * r7667605;
        double r7667607 = -3.554031892664371e-133;
        bool r7667608 = r7667600 <= r7667607;
        double r7667609 = a;
        double r7667610 = cbrt(r7667609);
        double r7667611 = r7667604 / r7667610;
        double r7667612 = r7667610 * r7667610;
        double r7667613 = r7667609 / r7667612;
        double r7667614 = r7667611 * r7667613;
        double r7667615 = r7667600 * r7667600;
        double r7667616 = r7667604 * r7667609;
        double r7667617 = r7667615 - r7667616;
        double r7667618 = sqrt(r7667617);
        double r7667619 = -r7667600;
        double r7667620 = r7667618 + r7667619;
        double r7667621 = r7667614 / r7667620;
        double r7667622 = 1.983916337927056e+89;
        bool r7667623 = r7667600 <= r7667622;
        double r7667624 = r7667600 / r7667609;
        double r7667625 = -r7667624;
        double r7667626 = r7667618 / r7667609;
        double r7667627 = r7667625 - r7667626;
        double r7667628 = 0.5;
        double r7667629 = r7667600 / r7667604;
        double r7667630 = r7667609 / r7667629;
        double r7667631 = -2.0;
        double r7667632 = r7667600 * r7667631;
        double r7667633 = fma(r7667628, r7667630, r7667632);
        double r7667634 = r7667633 / r7667609;
        double r7667635 = r7667623 ? r7667627 : r7667634;
        double r7667636 = r7667608 ? r7667621 : r7667635;
        double r7667637 = r7667602 ? r7667606 : r7667636;
        return r7667637;
}

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 < -6.473972066548491e+100

    1. Initial program 58.8

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

    3. Applied div-inv58.8

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

    4. Taylor expanded around -inf 2.3

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

    if -6.473972066548491e+100 < b_2 < -3.554031892664371e-133

    1. Initial program 39.0

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

    3. Applied div-inv39.1

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

    5. Applied flip--39.2

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

    6. Applied associate-*l/39.2

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

    7. Simplified13.1

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

    9. Applied add-cube-cbrt14.0

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

    10. Applied times-frac10.6

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

    if -3.554031892664371e-133 < b_2 < 1.983916337927056e+89

    1. Initial program 11.5

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

    3. Applied div-sub11.5

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

    if 1.983916337927056e+89 < b_2

    1. Initial program 42.0

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified4.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(-2 \cdot b_2\right)\right)}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{2}, \left(\frac{a}{\frac{b_2}{c}}\right), \left(b_2 \cdot -2\right)\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))