Average Error: 33.7 → 15.5
Time: 9.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.5407141653924318 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \sqrt{3} \cdot \left(\sqrt{3} \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\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 -2.2724541866372811 \cdot 10^{165}:\\
\;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \le 2.5407141653924318 \cdot 10^{-51}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \sqrt{3} \cdot \left(\sqrt{3} \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r143023 = b;
        double r143024 = -r143023;
        double r143025 = r143023 * r143023;
        double r143026 = 3.0;
        double r143027 = a;
        double r143028 = r143026 * r143027;
        double r143029 = c;
        double r143030 = r143028 * r143029;
        double r143031 = r143025 - r143030;
        double r143032 = sqrt(r143031);
        double r143033 = r143024 + r143032;
        double r143034 = r143033 / r143028;
        return r143034;
}
↓
double f(double a, double b, double c) {
        double r143035 = b;
        double r143036 = -2.272454186637281e+165;
        bool r143037 = r143035 <= r143036;
        double r143038 = 1.5;
        double r143039 = a;
        double r143040 = c;
        double r143041 = r143039 * r143040;
        double r143042 = r143041 / r143035;
        double r143043 = r143038 * r143042;
        double r143044 = r143043 - r143035;
        double r143045 = r143044 - r143035;
        double r143046 = 3.0;
        double r143047 = r143046 * r143039;
        double r143048 = r143045 / r143047;
        double r143049 = 2.5407141653924318e-51;
        bool r143050 = r143035 <= r143049;
        double r143051 = r143035 * r143035;
        double r143052 = sqrt(r143046);
        double r143053 = r143052 * r143041;
        double r143054 = r143052 * r143053;
        double r143055 = r143051 - r143054;
        double r143056 = sqrt(r143055);
        double r143057 = r143056 - r143035;
        double r143058 = r143057 / r143047;
        double r143059 = -1.5;
        double r143060 = r143059 * r143042;
        double r143061 = r143060 / r143047;
        double r143062 = r143050 ? r143058 : r143061;
        double r143063 = r143037 ? r143048 : r143062;
        return r143063;
}

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 3 regimes

  2. if b < -2.272454186637281e+165

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 10.5

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

    if -2.272454186637281e+165 < b < 2.5407141653924318e-51

    1. Initial program 13.5

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

    2. Simplified13.5

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

    4. Applied associate-*l*13.6

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

    6. Applied add-sqr-sqrt13.6

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

    7. Applied associate-*l*13.6

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

    if 2.5407141653924318e-51 < b

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded around inf 19.5

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

  4. Final simplification15.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \le 2.5407141653924318 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \sqrt{3} \cdot \left(\sqrt{3} \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))