Average Error: 28.9 → 16.2
Time: 14.2s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 1083.7283358723973:\\
\;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1358094 = b;
        double r1358095 = -r1358094;
        double r1358096 = r1358094 * r1358094;
        double r1358097 = 3.0;
        double r1358098 = a;
        double r1358099 = r1358097 * r1358098;
        double r1358100 = c;
        double r1358101 = r1358099 * r1358100;
        double r1358102 = r1358096 - r1358101;
        double r1358103 = sqrt(r1358102);
        double r1358104 = r1358095 + r1358103;
        double r1358105 = r1358104 / r1358099;
        return r1358105;
}
↓
double f(double a, double b, double c) {
        double r1358106 = b;
        double r1358107 = 1083.7283358723973;
        bool r1358108 = r1358106 <= r1358107;
        double r1358109 = -3.0;
        double r1358110 = a;
        double r1358111 = r1358109 * r1358110;
        double r1358112 = c;
        double r1358113 = r1358111 * r1358112;
        double r1358114 = r1358106 * r1358106;
        double r1358115 = r1358113 + r1358114;
        double r1358116 = sqrt(r1358115);
        double r1358117 = r1358115 * r1358116;
        double r1358118 = r1358106 * r1358114;
        double r1358119 = r1358117 - r1358118;
        double r1358120 = r1358106 * r1358116;
        double r1358121 = r1358114 + r1358120;
        double r1358122 = r1358115 + r1358121;
        double r1358123 = r1358119 / r1358122;
        double r1358124 = 3.0;
        double r1358125 = r1358110 * r1358124;
        double r1358126 = r1358123 / r1358125;
        double r1358127 = -0.5;
        double r1358128 = r1358112 / r1358106;
        double r1358129 = r1358127 * r1358128;
        double r1358130 = r1358108 ? r1358126 : r1358129;
        return r1358130;
}

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 < 1083.7283358723973

    1. Initial program 17.5

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

    2. Simplified17.5

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

    4. Applied flip3--17.6

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

    5. Simplified16.9

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

    6. Simplified16.9

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

    if 1083.7283358723973 < b

    1. Initial program 36.9

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

    2. Simplified36.9

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

    3. Taylor expanded around inf 15.8

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

    4. Taylor expanded around 0 15.7

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b} - b \cdot \left(b \cdot b\right)}{\left(\left(-3 \cdot a\right) \cdot c + b \cdot b\right) + \left(b \cdot b + b \cdot \sqrt{\left(-3 \cdot a\right) \cdot c + b \cdot b}\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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)))