Average Error: 34.5 → 10.1
Time: 4.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\
\;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r11560 = b_2;
        double r11561 = -r11560;
        double r11562 = r11560 * r11560;
        double r11563 = a;
        double r11564 = c;
        double r11565 = r11563 * r11564;
        double r11566 = r11562 - r11565;
        double r11567 = sqrt(r11566);
        double r11568 = r11561 - r11567;
        double r11569 = r11568 / r11563;
        return r11569;
}
↓
double f(double a, double b_2, double c) {
        double r11570 = b_2;
        double r11571 = -4.706781135059312e-92;
        bool r11572 = r11570 <= r11571;
        double r11573 = 1.0;
        double r11574 = -0.5;
        double r11575 = c;
        double r11576 = r11575 / r11570;
        double r11577 = r11574 * r11576;
        double r11578 = r11573 * r11577;
        double r11579 = 5.722235152988638e+98;
        bool r11580 = r11570 <= r11579;
        double r11581 = -r11570;
        double r11582 = r11570 * r11570;
        double r11583 = a;
        double r11584 = r11583 * r11575;
        double r11585 = r11582 - r11584;
        double r11586 = sqrt(r11585);
        double r11587 = r11581 - r11586;
        double r11588 = r11587 / r11583;
        double r11589 = r11573 * r11588;
        double r11590 = 0.5;
        double r11591 = r11590 * r11576;
        double r11592 = 2.0;
        double r11593 = r11570 / r11583;
        double r11594 = r11592 * r11593;
        double r11595 = r11591 - r11594;
        double r11596 = r11580 ? r11589 : r11595;
        double r11597 = r11572 ? r11578 : r11596;
        return r11597;
}

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 < -4.706781135059312e-92

    1. Initial program 52.4

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

    3. Applied div-inv52.4

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

    5. Applied *-un-lft-identity52.4

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

    6. Applied associate-*l*52.4

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

    7. Simplified52.4

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

    8. Taylor expanded around -inf 10.2

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

    if -4.706781135059312e-92 < b_2 < 5.722235152988638e+98

    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-lft-identity12.7

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

    6. Applied associate-*l*12.7

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

    7. Simplified12.6

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

    if 5.722235152988638e+98 < b_2

    1. Initial program 47.1

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))