Average Error: 33.9 → 13.4
Time: 19.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -182968668262934.46875:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{a}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -182968668262934.46875:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\
\;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{a}}{\left(-b_2\right) + \sqrt{{b_2}^{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 r42548 = b_2;
        double r42549 = -r42548;
        double r42550 = r42548 * r42548;
        double r42551 = a;
        double r42552 = c;
        double r42553 = r42551 * r42552;
        double r42554 = r42550 - r42553;
        double r42555 = sqrt(r42554);
        double r42556 = r42549 - r42555;
        double r42557 = r42556 / r42551;
        return r42557;
}
↓
double f(double a, double b_2, double c) {
        double r42558 = b_2;
        double r42559 = -182968668262934.47;
        bool r42560 = r42558 <= r42559;
        double r42561 = -0.5;
        double r42562 = c;
        double r42563 = r42562 / r42558;
        double r42564 = r42561 * r42563;
        double r42565 = 7.455592343308264e-170;
        bool r42566 = r42558 <= r42565;
        double r42567 = r42558 * r42558;
        double r42568 = r42567 - r42567;
        double r42569 = a;
        double r42570 = r42569 * r42562;
        double r42571 = r42568 + r42570;
        double r42572 = r42571 / r42569;
        double r42573 = -r42558;
        double r42574 = 2.0;
        double r42575 = pow(r42558, r42574);
        double r42576 = r42575 - r42570;
        double r42577 = sqrt(r42576);
        double r42578 = r42573 + r42577;
        double r42579 = r42572 / r42578;
        double r42580 = 0.5;
        double r42581 = r42580 * r42563;
        double r42582 = r42558 / r42569;
        double r42583 = r42574 * r42582;
        double r42584 = r42581 - r42583;
        double r42585 = r42566 ? r42579 : r42584;
        double r42586 = r42560 ? r42564 : r42585;
        return r42586;
}

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 < -182968668262934.47

    1. Initial program 56.0

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

    2. Taylor expanded around -inf 5.3

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

    if -182968668262934.47 < b_2 < 7.455592343308264e-170

    1. Initial program 24.3

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

    3. Applied clear-num24.3

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

    5. Applied div-inv24.4

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

    6. Applied add-cube-cbrt24.4

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

    7. Applied times-frac24.3

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

    8. Simplified24.3

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

    9. Simplified24.3

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(\left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c}\right)}\]
    10. Using strategy rm

    11. Applied flip--24.6

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c} \cdot \sqrt{{b_2}^{2} - a \cdot c}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}}\]

    12. Applied associate-*r/24.6

      \[\leadsto \color{blue}{\frac{\frac{1}{a} \cdot \left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{{b_2}^{2} - a \cdot c} \cdot \sqrt{{b_2}^{2} - a \cdot c}\right)}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}}\]

    13. Simplified17.3

      \[\leadsto \frac{\color{blue}{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{a}}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}\]

    if 7.455592343308264e-170 < b_2

    1. Initial program 23.0

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

    2. Taylor expanded around inf 17.1

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

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -182968668262934.46875:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 7.455592343308264166675918758902222662503 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{a}}{\left(-b_2\right) + \sqrt{{b_2}^{2} - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))