Average Error: 43.9 → 11.4
Time: 12.4s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.07798087007627403643983399206263129599392:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c}}\\ \end{array}\]
\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 0.07798087007627403643983399206263129599392:\\
\;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r69995 = b;
        double r69996 = -r69995;
        double r69997 = r69995 * r69995;
        double r69998 = 3.0;
        double r69999 = a;
        double r70000 = r69998 * r69999;
        double r70001 = c;
        double r70002 = r70000 * r70001;
        double r70003 = r69997 - r70002;
        double r70004 = sqrt(r70003);
        double r70005 = r69996 + r70004;
        double r70006 = r70005 / r70000;
        return r70006;
}
↓
double f(double a, double b, double c) {
        double r70007 = b;
        double r70008 = 0.07798087007627404;
        bool r70009 = r70007 <= r70008;
        double r70010 = r70007 * r70007;
        double r70011 = 3.0;
        double r70012 = a;
        double r70013 = r70011 * r70012;
        double r70014 = c;
        double r70015 = r70013 * r70014;
        double r70016 = fma(r70007, r70007, r70015);
        double r70017 = r70010 - r70016;
        double r70018 = r70010 - r70015;
        double r70019 = sqrt(r70018);
        double r70020 = r70007 + r70019;
        double r70021 = r70017 / r70020;
        double r70022 = r70021 / r70013;
        double r70023 = 1.0;
        double r70024 = -2.0;
        double r70025 = r70007 / r70014;
        double r70026 = r70024 * r70025;
        double r70027 = r70023 / r70026;
        double r70028 = r70009 ? r70022 : r70027;
        return r70028;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.07798087007627404

    1. Initial program 22.9

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

    2. Simplified22.9

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

    4. Applied flip--22.9

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

    5. Simplified22.2

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

    6. Simplified22.2

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

    if 0.07798087007627404 < b

    1. Initial program 47.1

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

    2. Simplified47.1

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

    3. Taylor expanded around inf 9.8

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
    4. Using strategy rm

    5. Applied clear-num9.9

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{-1.5 \cdot \frac{a \cdot c}{b}}}}\]

    6. Taylor expanded around 0 9.7

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.07798087007627403643983399206263129599392:\\ \;\;\;\;\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))