Average Error: 28.7 → 16.4
Time: 29.2s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + \left(a \cdot c\right) \cdot -3\right) \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b + \left(a \cdot c\right) \cdot -3\right)\right) + b \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]
\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 174.55343675894656:\\
\;\;\;\;\frac{\frac{\left(b \cdot b + \left(a \cdot c\right) \cdot -3\right) \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b + \left(a \cdot c\right) \cdot -3\right)\right) + b \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3}}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3706102 = b;
        double r3706103 = -r3706102;
        double r3706104 = r3706102 * r3706102;
        double r3706105 = 3.0;
        double r3706106 = a;
        double r3706107 = r3706105 * r3706106;
        double r3706108 = c;
        double r3706109 = r3706107 * r3706108;
        double r3706110 = r3706104 - r3706109;
        double r3706111 = sqrt(r3706110);
        double r3706112 = r3706103 + r3706111;
        double r3706113 = r3706112 / r3706107;
        return r3706113;
}
↓
double f(double a, double b, double c) {
        double r3706114 = b;
        double r3706115 = 174.55343675894656;
        bool r3706116 = r3706114 <= r3706115;
        double r3706117 = r3706114 * r3706114;
        double r3706118 = a;
        double r3706119 = c;
        double r3706120 = r3706118 * r3706119;
        double r3706121 = -3.0;
        double r3706122 = r3706120 * r3706121;
        double r3706123 = r3706117 + r3706122;
        double r3706124 = sqrt(r3706123);
        double r3706125 = r3706123 * r3706124;
        double r3706126 = r3706117 * r3706114;
        double r3706127 = r3706125 - r3706126;
        double r3706128 = r3706117 + r3706123;
        double r3706129 = r3706114 * r3706124;
        double r3706130 = r3706128 + r3706129;
        double r3706131 = r3706127 / r3706130;
        double r3706132 = 3.0;
        double r3706133 = r3706118 * r3706132;
        double r3706134 = r3706131 / r3706133;
        double r3706135 = r3706119 / r3706114;
        double r3706136 = -0.5;
        double r3706137 = r3706135 * r3706136;
        double r3706138 = r3706116 ? r3706134 : r3706137;
        return r3706138;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 174.55343675894656

    1. Initial program 15.8

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

    2. Simplified15.8

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

    4. Applied flip3--15.9

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

    5. Simplified15.2

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

    6. Simplified15.2

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

    if 174.55343675894656 < b

    1. Initial program 35.3

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

    2. Simplified35.3

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

    3. Taylor expanded around inf 17.1

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity17.1

      \[\leadsto \frac{\frac{-3}{2} \cdot \frac{a \cdot c}{\color{blue}{1 \cdot b}}}{3 \cdot a}\]

    6. Applied times-frac17.1

      \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{c}{b}\right)}}{3 \cdot a}\]

    7. Applied associate-*r*17.0

      \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot \frac{a}{1}\right) \cdot \frac{c}{b}}}{3 \cdot a}\]

    8. Simplified17.0

      \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot a\right)} \cdot \frac{c}{b}}{3 \cdot a}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\left(\frac{-3}{2} \cdot a\right) \cdot \frac{c}{b}}{\color{blue}{1 \cdot \left(3 \cdot a\right)}}\]

    11. Applied *-commutative17.0

      \[\leadsto \frac{\color{blue}{\frac{c}{b} \cdot \left(\frac{-3}{2} \cdot a\right)}}{1 \cdot \left(3 \cdot a\right)}\]

    12. Applied times-frac17.0

      \[\leadsto \color{blue}{\frac{\frac{c}{b}}{1} \cdot \frac{\frac{-3}{2} \cdot a}{3 \cdot a}}\]

    13. Simplified17.0

      \[\leadsto \color{blue}{\frac{c}{b}} \cdot \frac{\frac{-3}{2} \cdot a}{3 \cdot a}\]

    14. Simplified17.0

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

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\left(b \cdot b + \left(a \cdot c\right) \cdot -3\right) \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b + \left(a \cdot c\right) \cdot -3\right)\right) + b \cdot \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :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)))