Average Error: 52.4 → 0.2
Time: 18.3s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\left(-a\right) \cdot c}{a}}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot c, 3, b \cdot b\right)} + b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\left(-a\right) \cdot c}{a}}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot c, 3, b \cdot b\right)} + b}
double f(double a, double b, double c) {
        double r114973 = b;
        double r114974 = -r114973;
        double r114975 = r114973 * r114973;
        double r114976 = 3.0;
        double r114977 = a;
        double r114978 = r114976 * r114977;
        double r114979 = c;
        double r114980 = r114978 * r114979;
        double r114981 = r114975 - r114980;
        double r114982 = sqrt(r114981);
        double r114983 = r114974 + r114982;
        double r114984 = r114983 / r114978;
        return r114984;
}
↓
double f(double a, double b, double c) {
        double r114985 = a;
        double r114986 = -r114985;
        double r114987 = c;
        double r114988 = r114986 * r114987;
        double r114989 = r114988 / r114985;
        double r114990 = 3.0;
        double r114991 = b;
        double r114992 = r114991 * r114991;
        double r114993 = fma(r114988, r114990, r114992);
        double r114994 = sqrt(r114993);
        double r114995 = r114994 + r114991;
        double r114996 = r114989 / r114995;
        return r114996;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.4

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

  2. Simplified52.4

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

  4. Applied flip--52.4

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied div-inv0.6

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Simplified0.5

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

  13. Applied distribute-frac-neg0.5

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

  14. Applied distribute-lft-neg-out0.5

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))