Average Error: 44.3 → 0.3
Time: 16.3s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a \cdot \left(-3\right), c, {b}^{2}\right)}}{c} \cdot \frac{3 \cdot a}{a \cdot \left(-3\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a \cdot \left(-3\right), c, {b}^{2}\right)}}{c} \cdot \frac{3 \cdot a}{a \cdot \left(-3\right)}}
double f(double a, double b, double c) {
        double r132307 = b;
        double r132308 = -r132307;
        double r132309 = r132307 * r132307;
        double r132310 = 3.0;
        double r132311 = a;
        double r132312 = r132310 * r132311;
        double r132313 = c;
        double r132314 = r132312 * r132313;
        double r132315 = r132309 - r132314;
        double r132316 = sqrt(r132315);
        double r132317 = r132308 + r132316;
        double r132318 = r132317 / r132312;
        return r132318;
}
↓
double f(double a, double b, double c) {
        double r132319 = 1.0;
        double r132320 = b;
        double r132321 = a;
        double r132322 = 3.0;
        double r132323 = -r132322;
        double r132324 = r132321 * r132323;
        double r132325 = c;
        double r132326 = 2.0;
        double r132327 = pow(r132320, r132326);
        double r132328 = fma(r132324, r132325, r132327);
        double r132329 = sqrt(r132328);
        double r132330 = r132320 + r132329;
        double r132331 = r132330 / r132325;
        double r132332 = r132322 * r132321;
        double r132333 = r132332 / r132324;
        double r132334 = r132331 * r132333;
        double r132335 = r132319 / r132334;
        return r132335;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 44.3

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified44.3

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

  4. Applied flip--44.2

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

  5. Simplified0.6

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

  6. Simplified0.6

    \[\leadsto \frac{\frac{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}{\color{blue}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied clear-num0.6

    \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{\mathsf{fma}\left(-c \cdot a, 3, 0\right)}{b + \sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}}}}}\]

  9. Simplified0.3

    \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 3}{\left(-3\right) \cdot a} \cdot \frac{b + \sqrt{\mathsf{fma}\left(\left(-3\right) \cdot a, c, {b}^{2}\right)}}{c}}}\]

  10. Final simplification0.3

    \[\leadsto \frac{1}{\frac{b + \sqrt{\mathsf{fma}\left(a \cdot \left(-3\right), c, {b}^{2}\right)}}{c} \cdot \frac{3 \cdot a}{a \cdot \left(-3\right)}}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))