Average Error: 43.2 → 0.2
Time: 19.1s
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 r71595 = b;
        double r71596 = -r71595;
        double r71597 = r71595 * r71595;
        double r71598 = 3.0;
        double r71599 = a;
        double r71600 = r71598 * r71599;
        double r71601 = c;
        double r71602 = r71600 * r71601;
        double r71603 = r71597 - r71602;
        double r71604 = sqrt(r71603);
        double r71605 = r71596 + r71604;
        double r71606 = r71605 / r71600;
        return r71606;
}
↓
double f(double a, double b, double c) {
        double r71607 = -3.0;
        double r71608 = a;
        double r71609 = r71607 * r71608;
        double r71610 = 3.0;
        double r71611 = r71610 * r71608;
        double r71612 = c;
        double r71613 = -r71611;
        double r71614 = b;
        double r71615 = r71614 * r71614;
        double r71616 = fma(r71612, r71613, r71615);
        double r71617 = sqrt(r71616);
        double r71618 = r71617 + r71614;
        double r71619 = r71612 / r71618;
        double r71620 = r71611 / r71619;
        double r71621 = r71609 / r71620;
        return r71621;
}

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)))