Average Error: 52.1 → 6.1
Time: 4.1s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -219230968.68811727:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -219230968.68811727:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r120016 = b;
        double r120017 = -r120016;
        double r120018 = r120016 * r120016;
        double r120019 = 3.0;
        double r120020 = a;
        double r120021 = r120019 * r120020;
        double r120022 = c;
        double r120023 = r120021 * r120022;
        double r120024 = r120018 - r120023;
        double r120025 = sqrt(r120024);
        double r120026 = r120017 + r120025;
        double r120027 = r120026 / r120021;
        return r120027;
}
↓
double f(double a, double b, double c) {
        double r120028 = b;
        double r120029 = -r120028;
        double r120030 = r120028 * r120028;
        double r120031 = 3.0;
        double r120032 = a;
        double r120033 = r120031 * r120032;
        double r120034 = c;
        double r120035 = r120033 * r120034;
        double r120036 = r120030 - r120035;
        double r120037 = sqrt(r120036);
        double r120038 = r120029 + r120037;
        double r120039 = r120038 / r120033;
        double r120040 = -219230968.68811727;
        bool r120041 = r120039 <= r120040;
        double r120042 = -r120036;
        double r120043 = fma(r120028, r120028, r120042);
        double r120044 = r120029 - r120037;
        double r120045 = r120043 / r120044;
        double r120046 = r120045 / r120033;
        double r120047 = -0.5;
        double r120048 = r120034 / r120028;
        double r120049 = r120047 * r120048;
        double r120050 = r120041 ? r120046 : r120049;
        return r120050;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) < -219230968.68811727

    1. Initial program 20.7

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

    3. Applied flip-+20.7

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

    4. Simplified19.7

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

    if -219230968.68811727 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

    1. Initial program 53.7

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

    2. Taylor expanded around inf 5.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -219230968.68811727:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))