Average Error: 43.3 → 0.5
Time: 20.6s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(4 \cdot a\right) \cdot c}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\left(4 \cdot a\right) \cdot c}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}{a}
double f(double a, double b, double c) {
        double r46990 = b;
        double r46991 = -r46990;
        double r46992 = r46990 * r46990;
        double r46993 = 4.0;
        double r46994 = a;
        double r46995 = r46993 * r46994;
        double r46996 = c;
        double r46997 = r46995 * r46996;
        double r46998 = r46992 - r46997;
        double r46999 = sqrt(r46998);
        double r47000 = r46991 + r46999;
        double r47001 = 2.0;
        double r47002 = r47001 * r46994;
        double r47003 = r47000 / r47002;
        return r47003;
}
↓
double f(double a, double b, double c) {
        double r47004 = 4.0;
        double r47005 = a;
        double r47006 = r47004 * r47005;
        double r47007 = c;
        double r47008 = r47006 * r47007;
        double r47009 = 2.0;
        double r47010 = r47008 / r47009;
        double r47011 = 1.0;
        double r47012 = b;
        double r47013 = -r47012;
        double r47014 = 4.0;
        double r47015 = pow(r47012, r47014);
        double r47016 = r47008 * r47008;
        double r47017 = r47015 - r47016;
        double r47018 = r47005 * r47007;
        double r47019 = r47018 * r47004;
        double r47020 = fma(r47012, r47012, r47019);
        double r47021 = r47017 / r47020;
        double r47022 = sqrt(r47021);
        double r47023 = r47013 - r47022;
        double r47024 = r47011 / r47023;
        double r47025 = r47024 / r47005;
        double r47026 = r47010 * r47025;
        return r47026;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.3

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

  3. Applied flip-+43.4

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

  4. Simplified0.4

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

  6. Applied flip--0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  10. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(0 + \left(4 \cdot a\right) \cdot c\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}}{2 \cdot a}\]

  11. Applied times-frac0.5

    \[\leadsto \color{blue}{\frac{0 + \left(4 \cdot a\right) \cdot c}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}{a}}\]

  12. Simplified0.5

    \[\leadsto \color{blue}{\frac{\left(4 \cdot a\right) \cdot c}{2}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}{a}\]

  13. Final simplification0.5

    \[\leadsto \frac{\left(4 \cdot a\right) \cdot c}{2} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot 4\right)}}}}{a}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))