Average Error: 34.1 → 6.6
Time: 18.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_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 r78578 = b_2;
        double r78579 = -r78578;
        double r78580 = r78578 * r78578;
        double r78581 = a;
        double r78582 = c;
        double r78583 = r78581 * r78582;
        double r78584 = r78580 - r78583;
        double r78585 = sqrt(r78584);
        double r78586 = r78579 - r78585;
        double r78587 = r78586 / r78581;
        return r78587;
}
↓
double f(double a, double b_2, double c) {
        double r78588 = b_2;
        double r78589 = -3.5132588248780117e+152;
        bool r78590 = r78588 <= r78589;
        double r78591 = -0.5;
        double r78592 = c;
        double r78593 = r78592 / r78588;
        double r78594 = r78591 * r78593;
        double r78595 = 8.216265756828381e-276;
        bool r78596 = r78588 <= r78595;
        double r78597 = r78588 * r78588;
        double r78598 = a;
        double r78599 = r78598 * r78592;
        double r78600 = r78597 - r78599;
        double r78601 = sqrt(r78600);
        double r78602 = r78601 - r78588;
        double r78603 = r78592 / r78602;
        double r78604 = 5.031608061939103e+53;
        bool r78605 = r78588 <= r78604;
        double r78606 = 1.0;
        double r78607 = -r78588;
        double r78608 = r78607 - r78601;
        double r78609 = r78598 / r78608;
        double r78610 = r78606 / r78609;
        double r78611 = 0.5;
        double r78612 = r78611 * r78593;
        double r78613 = 2.0;
        double r78614 = r78588 / r78598;
        double r78615 = r78613 * r78614;
        double r78616 = r78612 - r78615;
        double r78617 = r78605 ? r78610 : r78616;
        double r78618 = r78596 ? r78603 : r78617;
        double r78619 = r78590 ? r78594 : r78618;
        return r78619;
}

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 < -3.5132588248780117e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.4

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

    if -3.5132588248780117e+152 < b_2 < 8.216265756828381e-276

    1. Initial program 33.1

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

    3. Applied flip--33.1

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

    4. Simplified15.2

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

    5. Simplified15.2

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

    7. Applied *-un-lft-identity15.2

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

    8. Applied times-frac14.7

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

    9. Applied associate-/l*10.5

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

    10. Simplified7.8

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

    if 8.216265756828381e-276 < b_2 < 5.031608061939103e+53

    1. Initial program 9.3

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

    3. Applied clear-num9.5

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

    if 5.031608061939103e+53 < b_2

    1. Initial program 39.6

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

    2. Taylor expanded around inf 5.7

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.513258824878011748257049801344805265531 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 8.216265756828381163830890149037103205802 \cdot 10^{-276}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 5.031608061939102936286074782173578716838 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))