Average Error: 43.2 → 0.2
Time: 18.5s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-3 \cdot a}{\frac{3 \cdot a}{\frac{c}{\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} + b}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{-3 \cdot a}{\frac{3 \cdot a}{\frac{c}{\sqrt{\mathsf{fma}\left(c, -3 \cdot a, b \cdot b\right)} + b}}}
double f(double a, double b, double c) {
        double r141397 = b;
        double r141398 = -r141397;
        double r141399 = r141397 * r141397;
        double r141400 = 3.0;
        double r141401 = a;
        double r141402 = r141400 * r141401;
        double r141403 = c;
        double r141404 = r141402 * r141403;
        double r141405 = r141399 - r141404;
        double r141406 = sqrt(r141405);
        double r141407 = r141398 + r141406;
        double r141408 = r141407 / r141402;
        return r141408;
}
↓
double f(double a, double b, double c) {
        double r141409 = -3.0;
        double r141410 = a;
        double r141411 = r141409 * r141410;
        double r141412 = 3.0;
        double r141413 = r141412 * r141410;
        double r141414 = c;
        double r141415 = -r141413;
        double r141416 = b;
        double r141417 = r141416 * r141416;
        double r141418 = fma(r141414, r141415, r141417);
        double r141419 = sqrt(r141418);
        double r141420 = r141419 + r141416;
        double r141421 = r141414 / r141420;
        double r141422 = r141413 / r141421;
        double r141423 = r141411 / r141422;
        return r141423;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.2

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

  2. Simplified43.2

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

  4. Applied flip--43.2

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

  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. Taylor expanded around 0 0.6

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied times-frac0.3

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

  12. Applied associate-/l*0.2

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +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)))