Average Error: 33.7 → 8.4
Time: 20.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r2777404 = b_2;
        double r2777405 = -r2777404;
        double r2777406 = r2777404 * r2777404;
        double r2777407 = a;
        double r2777408 = c;
        double r2777409 = r2777407 * r2777408;
        double r2777410 = r2777406 - r2777409;
        double r2777411 = sqrt(r2777410);
        double r2777412 = r2777405 - r2777411;
        double r2777413 = r2777412 / r2777407;
        return r2777413;
}
↓
double f(double a, double b_2, double c) {
        double r2777414 = b_2;
        double r2777415 = -3.234164035284793e+22;
        bool r2777416 = r2777414 <= r2777415;
        double r2777417 = -0.5;
        double r2777418 = c;
        double r2777419 = r2777418 / r2777414;
        double r2777420 = r2777417 * r2777419;
        double r2777421 = -6.3209183644448e-115;
        bool r2777422 = r2777414 <= r2777421;
        double r2777423 = a;
        double r2777424 = r2777418 * r2777423;
        double r2777425 = r2777414 * r2777414;
        double r2777426 = r2777425 - r2777424;
        double r2777427 = sqrt(r2777426);
        double r2777428 = r2777427 - r2777414;
        double r2777429 = r2777424 / r2777428;
        double r2777430 = r2777429 / r2777423;
        double r2777431 = 2.026128983134594e+103;
        bool r2777432 = r2777414 <= r2777431;
        double r2777433 = 1.0;
        double r2777434 = r2777433 / r2777423;
        double r2777435 = -r2777414;
        double r2777436 = r2777435 - r2777427;
        double r2777437 = r2777434 * r2777436;
        double r2777438 = 0.5;
        double r2777439 = r2777414 / r2777423;
        double r2777440 = -2.0;
        double r2777441 = r2777439 * r2777440;
        double r2777442 = fma(r2777438, r2777419, r2777441);
        double r2777443 = r2777432 ? r2777437 : r2777442;
        double r2777444 = r2777422 ? r2777430 : r2777443;
        double r2777445 = r2777416 ? r2777420 : r2777444;
        return r2777445;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.234164035284793e+22

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 4.6

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

    if -3.234164035284793e+22 < b_2 < -6.3209183644448e-115

    1. Initial program 38.3

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

    3. Applied flip--38.4

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

    4. Simplified15.5

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified15.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -6.3209183644448e-115 < b_2 < 2.026128983134594e+103

    1. Initial program 11.3

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

    3. Applied div-inv11.4

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

    if 2.026128983134594e+103 < b_2

    1. Initial program 45.2

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

    2. Taylor expanded around inf 3.2

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

    3. Simplified3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.234164035284793 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -6.3209183644448 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.026128983134594 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))