Average Error: 34.0 → 9.6
Time: 4.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.0366436397824178 \cdot 10^{68}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -8.21218726880377109 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{elif}\;b_2 \le -4.05237835825691163 \cdot 10^{-102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\ \;\;\;\;\frac{\left(-b_2\right) + \left(-\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.0366436397824178 \cdot 10^{68}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -8.21218726880377109 \cdot 10^{-92}:\\
\;\;\;\;\frac{\frac{\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

\mathbf{elif}\;b_2 \le -4.05237835825691163 \cdot 10^{-102}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\
\;\;\;\;\frac{\left(-b_2\right) + \left(-\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\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 r100540 = b_2;
        double r100541 = -r100540;
        double r100542 = r100540 * r100540;
        double r100543 = a;
        double r100544 = c;
        double r100545 = r100543 * r100544;
        double r100546 = r100542 - r100545;
        double r100547 = sqrt(r100546);
        double r100548 = r100541 - r100547;
        double r100549 = r100548 / r100543;
        return r100549;
}
↓
double f(double a, double b_2, double c) {
        double r100550 = b_2;
        double r100551 = -1.0366436397824178e+68;
        bool r100552 = r100550 <= r100551;
        double r100553 = -0.5;
        double r100554 = c;
        double r100555 = r100554 / r100550;
        double r100556 = r100553 * r100555;
        double r100557 = -8.212187268803771e-92;
        bool r100558 = r100550 <= r100557;
        double r100559 = 2.0;
        double r100560 = pow(r100550, r100559);
        double r100561 = r100560 - r100560;
        double r100562 = a;
        double r100563 = r100562 * r100554;
        double r100564 = r100561 + r100563;
        double r100565 = -r100550;
        double r100566 = r100550 * r100550;
        double r100567 = r100566 - r100563;
        double r100568 = sqrt(r100567);
        double r100569 = r100565 + r100568;
        double r100570 = r100564 / r100569;
        double r100571 = r100570 / r100562;
        double r100572 = -4.052378358256912e-102;
        bool r100573 = r100550 <= r100572;
        double r100574 = 5.349311795482947e+30;
        bool r100575 = r100550 <= r100574;
        double r100576 = -r100568;
        double r100577 = r100565 + r100576;
        double r100578 = r100577 / r100562;
        double r100579 = 0.5;
        double r100580 = r100579 * r100555;
        double r100581 = r100550 / r100562;
        double r100582 = r100559 * r100581;
        double r100583 = r100580 - r100582;
        double r100584 = r100575 ? r100578 : r100583;
        double r100585 = r100573 ? r100556 : r100584;
        double r100586 = r100558 ? r100571 : r100585;
        double r100587 = r100552 ? r100556 : r100586;
        return r100587;
}

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 < -1.0366436397824178e+68 or -8.212187268803771e-92 < b_2 < -4.052378358256912e-102

    1. Initial program 57.1

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

    2. Taylor expanded around -inf 4.6

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

    if -1.0366436397824178e+68 < b_2 < -8.212187268803771e-92

    1. Initial program 42.4

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

    3. Applied add-cube-cbrt44.7

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

    4. Applied fma-neg45.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]
    5. Using strategy rm

    6. Applied fma-udef44.7

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

    7. Simplified42.4

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

    9. Applied flip-+42.4

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

    10. Simplified15.8

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

    11. Simplified15.8

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

    if -4.052378358256912e-102 < b_2 < 5.349311795482947e+30

    1. Initial program 13.0

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied fma-neg13.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}, \sqrt[3]{-b_2}, -\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]
    5. Using strategy rm

    6. Applied fma-udef13.1

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

    7. Simplified13.0

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

    if 5.349311795482947e+30 < b_2

    1. Initial program 35.5

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

    2. Taylor expanded around inf 6.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.0366436397824178 \cdot 10^{68}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -8.21218726880377109 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{elif}\;b_2 \le -4.05237835825691163 \cdot 10^{-102}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.34931179548294658 \cdot 10^{30}:\\ \;\;\;\;\frac{\left(-b_2\right) + \left(-\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))