Average Error: 34.6 → 9.9
Time: 17.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array}\]
\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 -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\

\mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-0.5 \cdot c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r159771 = b;
        double r159772 = -r159771;
        double r159773 = r159771 * r159771;
        double r159774 = 3.0;
        double r159775 = a;
        double r159776 = r159774 * r159775;
        double r159777 = c;
        double r159778 = r159776 * r159777;
        double r159779 = r159773 - r159778;
        double r159780 = sqrt(r159779);
        double r159781 = r159772 + r159780;
        double r159782 = r159781 / r159776;
        return r159782;
}
↓
double f(double a, double b, double c) {
        double r159783 = b;
        double r159784 = -7.943482039519134e+75;
        bool r159785 = r159783 <= r159784;
        double r159786 = -2.0;
        double r159787 = a;
        double r159788 = 1.5;
        double r159789 = r159787 * r159788;
        double r159790 = c;
        double r159791 = r159783 / r159790;
        double r159792 = r159789 / r159791;
        double r159793 = fma(r159786, r159783, r159792);
        double r159794 = 3.0;
        double r159795 = r159793 / r159794;
        double r159796 = r159795 / r159787;
        double r159797 = 8.08526583505735e-63;
        bool r159798 = r159783 <= r159797;
        double r159799 = r159794 * r159787;
        double r159800 = -r159790;
        double r159801 = r159799 * r159800;
        double r159802 = fma(r159783, r159783, r159801);
        double r159803 = sqrt(r159802);
        double r159804 = r159803 - r159783;
        double r159805 = r159804 / r159794;
        double r159806 = r159805 / r159787;
        double r159807 = -0.5;
        double r159808 = r159807 * r159790;
        double r159809 = r159808 / r159783;
        double r159810 = r159798 ? r159806 : r159809;
        double r159811 = r159785 ? r159796 : r159810;
        return r159811;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -7.943482039519134e+75

    1. Initial program 42.8

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

    3. Applied associate-/r*42.8

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

    4. Simplified42.8

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

    5. Taylor expanded around -inf 9.7

      \[\leadsto \frac{\frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3}}{a}\]

    6. Simplified4.5

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}}{3}}{a}\]

    if -7.943482039519134e+75 < b < 8.08526583505735e-63

    1. Initial program 13.7

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

    3. Applied associate-/r*13.7

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

    4. Simplified13.7

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

    if 8.08526583505735e-63 < b

    1. Initial program 53.6

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

    2. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

    3. Simplified8.3

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, b, \frac{a \cdot 1.5}{\frac{b}{c}}\right)}{3}}{a}\\ \mathbf{elif}\;b \le 8.085265835057349842233247168077451568119 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot \left(-c\right)\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))