Average Error: 28.7 → 17.3
Time: 8.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(3 \cdot a\right)\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{3}}{a} \cdot \sqrt{a}} \cdot \frac{-1.5}{\frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 17.714642954298647:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(3 \cdot a\right)\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{3}}{a} \cdot \sqrt{a}} \cdot \frac{-1.5}{\frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}\\

\end{array}
double f(double a, double b, double c) {
        double r145986 = b;
        double r145987 = -r145986;
        double r145988 = r145986 * r145986;
        double r145989 = 3.0;
        double r145990 = a;
        double r145991 = r145989 * r145990;
        double r145992 = c;
        double r145993 = r145991 * r145992;
        double r145994 = r145988 - r145993;
        double r145995 = sqrt(r145994);
        double r145996 = r145987 + r145995;
        double r145997 = r145996 / r145991;
        return r145997;
}
↓
double f(double a, double b, double c) {
        double r145998 = b;
        double r145999 = 17.714642954298647;
        bool r146000 = r145998 <= r145999;
        double r146001 = r145998 * r145998;
        double r146002 = c;
        double r146003 = 3.0;
        double r146004 = a;
        double r146005 = r146003 * r146004;
        double r146006 = r146002 * r146005;
        double r146007 = fma(r145998, r145998, r146006);
        double r146008 = r146001 - r146007;
        double r146009 = r146005 * r146002;
        double r146010 = r146001 - r146009;
        double r146011 = sqrt(r146010);
        double r146012 = r146011 + r145998;
        double r146013 = r146008 / r146012;
        double r146014 = r146013 / r146005;
        double r146015 = 1.0;
        double r146016 = sqrt(r146003);
        double r146017 = r146016 / r146004;
        double r146018 = sqrt(r146004);
        double r146019 = r146017 * r146018;
        double r146020 = r146015 / r146019;
        double r146021 = -1.5;
        double r146022 = r146002 / r145998;
        double r146023 = r146022 / r146018;
        double r146024 = r146016 / r146023;
        double r146025 = r146021 / r146024;
        double r146026 = r146020 * r146025;
        double r146027 = r146000 ? r146014 : r146026;
        return r146027;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 17.714642954298647

    1. Initial program 14.0

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

    2. Simplified14.0

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

    4. Applied flip--14.0

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

    5. Simplified13.4

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

    if 17.714642954298647 < b

    1. Initial program 33.6

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

    2. Simplified33.6

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

    3. Taylor expanded around inf 18.5

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied associate-/l*18.5

      \[\leadsto \color{blue}{\frac{-1.5}{\frac{3 \cdot a}{\frac{a \cdot c}{b}}}}\]

    6. Simplified18.5

      \[\leadsto \frac{-1.5}{\color{blue}{\frac{3}{\frac{\frac{a \cdot c}{b}}{a}}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt18.5

      \[\leadsto \frac{-1.5}{\frac{3}{\frac{\frac{a \cdot c}{b}}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}}}}\]

    9. Applied *-un-lft-identity18.5

      \[\leadsto \frac{-1.5}{\frac{3}{\frac{\frac{a \cdot c}{\color{blue}{1 \cdot b}}}{\sqrt{a} \cdot \sqrt{a}}}}\]

    10. Applied times-frac18.5

      \[\leadsto \frac{-1.5}{\frac{3}{\frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{b}}}{\sqrt{a} \cdot \sqrt{a}}}}\]

    11. Applied times-frac18.5

      \[\leadsto \frac{-1.5}{\frac{3}{\color{blue}{\frac{\frac{a}{1}}{\sqrt{a}} \cdot \frac{\frac{c}{b}}{\sqrt{a}}}}}\]

    12. Applied add-sqr-sqrt18.6

      \[\leadsto \frac{-1.5}{\frac{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}{\frac{\frac{a}{1}}{\sqrt{a}} \cdot \frac{\frac{c}{b}}{\sqrt{a}}}}\]

    13. Applied times-frac18.5

      \[\leadsto \frac{-1.5}{\color{blue}{\frac{\sqrt{3}}{\frac{\frac{a}{1}}{\sqrt{a}}} \cdot \frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}}\]

    14. Applied *-un-lft-identity18.5

      \[\leadsto \frac{\color{blue}{1 \cdot -1.5}}{\frac{\sqrt{3}}{\frac{\frac{a}{1}}{\sqrt{a}}} \cdot \frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}\]

    15. Applied times-frac18.6

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{\frac{\frac{a}{1}}{\sqrt{a}}}} \cdot \frac{-1.5}{\frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}}\]

    16. Simplified18.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{3}}{a} \cdot \sqrt{a}}} \cdot \frac{-1.5}{\frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 17.714642954298647:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(3 \cdot a\right)\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{3}}{a} \cdot \sqrt{a}} \cdot \frac{-1.5}{\frac{\sqrt{3}}{\frac{\frac{c}{b}}{\sqrt{a}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))