Average Error: 43.1 → 0.4
Time: 23.0s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-4\right), c, 0\right)}{b + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(4, c \cdot \left(-a\right), {b}^{2}\right)\right)}^{3}}}}}{a}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-4\right), c, 0\right)}{b + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(4, c \cdot \left(-a\right), {b}^{2}\right)\right)}^{3}}}}}{a}}{2}
double f(double a, double b, double c) {
        double r58355 = b;
        double r58356 = -r58355;
        double r58357 = r58355 * r58355;
        double r58358 = 4.0;
        double r58359 = a;
        double r58360 = r58358 * r58359;
        double r58361 = c;
        double r58362 = r58360 * r58361;
        double r58363 = r58357 - r58362;
        double r58364 = sqrt(r58363);
        double r58365 = r58356 + r58364;
        double r58366 = 2.0;
        double r58367 = r58366 * r58359;
        double r58368 = r58365 / r58367;
        return r58368;
}
↓
double f(double a, double b, double c) {
        double r58369 = a;
        double r58370 = 4.0;
        double r58371 = -r58370;
        double r58372 = r58369 * r58371;
        double r58373 = c;
        double r58374 = 0.0;
        double r58375 = fma(r58372, r58373, r58374);
        double r58376 = b;
        double r58377 = -r58369;
        double r58378 = r58373 * r58377;
        double r58379 = 2.0;
        double r58380 = pow(r58376, r58379);
        double r58381 = fma(r58370, r58378, r58380);
        double r58382 = 3.0;
        double r58383 = pow(r58381, r58382);
        double r58384 = cbrt(r58383);
        double r58385 = sqrt(r58384);
        double r58386 = r58376 + r58385;
        double r58387 = r58375 / r58386;
        double r58388 = r58387 / r58369;
        double r58389 = 2.0;
        double r58390 = r58388 / r58389;
        return r58390;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.1

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified43.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}{2}}\]
  3. Using strategy rm

  4. Applied flip--43.1

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}}{a}}{2}\]

  5. Simplified0.4

    \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(-a \cdot 4, c, 0\right)}}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}{a}}{2}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-a \cdot 4, c, 0\right)}{\color{blue}{b + \sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}}}}{a}}{2}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.4

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-a \cdot 4, c, 0\right)}{b + \sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right) \cdot \mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)\right) \cdot \mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}}}}}{a}}{2}\]

  9. Simplified0.4

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(-a \cdot 4, c, 0\right)}{b + \sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(4, \left(-c\right) \cdot a, {b}^{2}\right)\right)}^{3}}}}}}{a}}{2}\]

  10. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-4\right), c, 0\right)}{b + \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(4, c \cdot \left(-a\right), {b}^{2}\right)\right)}^{3}}}}}{a}}{2}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))