Average Error: 34.5 → 10.1
Time: 4.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;1 \cdot \left(\frac{-1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]
\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}:\\
\;\;\;\;1 \cdot \left(\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r11586 = b_2;
        double r11587 = -r11586;
        double r11588 = r11586 * r11586;
        double r11589 = a;
        double r11590 = c;
        double r11591 = r11589 * r11590;
        double r11592 = r11588 - r11591;
        double r11593 = sqrt(r11592);
        double r11594 = r11587 - r11593;
        double r11595 = r11594 / r11589;
        return r11595;
}
↓
double f(double a, double b_2, double c) {
        double r11596 = b_2;
        double r11597 = -4.706781135059312e-92;
        bool r11598 = r11596 <= r11597;
        double r11599 = 1.0;
        double r11600 = -0.5;
        double r11601 = c;
        double r11602 = r11601 / r11596;
        double r11603 = r11600 * r11602;
        double r11604 = r11599 * r11603;
        double r11605 = 5.722235152988638e+98;
        bool r11606 = r11596 <= r11605;
        double r11607 = -r11596;
        double r11608 = r11596 * r11596;
        double r11609 = a;
        double r11610 = r11609 * r11601;
        double r11611 = r11608 - r11610;
        double r11612 = sqrt(r11611);
        double r11613 = r11607 - r11612;
        double r11614 = r11613 / r11609;
        double r11615 = r11599 * r11614;
        double r11616 = 0.5;
        double r11617 = r11616 * r11602;
        double r11618 = 2.0;
        double r11619 = r11596 / r11609;
        double r11620 = r11618 * r11619;
        double r11621 = r11617 - r11620;
        double r11622 = r11599 * r11621;
        double r11623 = r11606 ? r11615 : r11622;
        double r11624 = r11598 ? r11604 : r11623;
        return r11624;
}

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. Using strategy rm

    3. Applied div-inv47.1

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

    5. Applied *-un-lft-identity47.1

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

    6. Applied associate-*l*47.1

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

    7. Simplified47.1

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

    8. Taylor expanded around inf 3.6

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

  4. Final simplification10.1

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

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))