Average Error: 43.5 → 11.4
Time: 14.2s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.0012535322255036849:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3703102 = b;
        double r3703103 = -r3703102;
        double r3703104 = r3703102 * r3703102;
        double r3703105 = 3.0;
        double r3703106 = a;
        double r3703107 = r3703105 * r3703106;
        double r3703108 = c;
        double r3703109 = r3703107 * r3703108;
        double r3703110 = r3703104 - r3703109;
        double r3703111 = sqrt(r3703110);
        double r3703112 = r3703103 + r3703111;
        double r3703113 = r3703112 / r3703107;
        return r3703113;
}
↓
double f(double a, double b, double c) {
        double r3703114 = b;
        double r3703115 = 0.0012535322255036849;
        bool r3703116 = r3703114 <= r3703115;
        double r3703117 = r3703114 * r3703114;
        double r3703118 = c;
        double r3703119 = 3.0;
        double r3703120 = a;
        double r3703121 = r3703119 * r3703120;
        double r3703122 = r3703118 * r3703121;
        double r3703123 = r3703117 - r3703122;
        double r3703124 = sqrt(r3703123);
        double r3703125 = r3703123 * r3703124;
        double r3703126 = r3703117 * r3703114;
        double r3703127 = r3703125 - r3703126;
        double r3703128 = r3703114 * r3703124;
        double r3703129 = r3703128 + r3703117;
        double r3703130 = r3703123 + r3703129;
        double r3703131 = r3703127 / r3703130;
        double r3703132 = r3703131 / r3703121;
        double r3703133 = -0.5;
        double r3703134 = r3703118 / r3703114;
        double r3703135 = r3703133 * r3703134;
        double r3703136 = r3703116 ? r3703132 : r3703135;
        return r3703136;
}

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 < 0.0012535322255036849

    1. Initial program 20.3

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

    2. Simplified20.3

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

    4. Applied flip3--20.5

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

    5. Simplified19.9

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

    6. Simplified19.9

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

    if 0.0012535322255036849 < b

    1. Initial program 45.8

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

    2. Simplified45.8

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

    3. Taylor expanded around inf 10.6

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0012535322255036849:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))