Average Error: 44.0 → 10.1
Time: 4.4s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -3.24447403724483937 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.24447403724483937 \cdot 10^{-11}:\\
\;\;\;\;\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 r68598 = b;
        double r68599 = -r68598;
        double r68600 = r68598 * r68598;
        double r68601 = 3.0;
        double r68602 = a;
        double r68603 = r68601 * r68602;
        double r68604 = c;
        double r68605 = r68603 * r68604;
        double r68606 = r68600 - r68605;
        double r68607 = sqrt(r68606);
        double r68608 = r68599 + r68607;
        double r68609 = r68608 / r68603;
        return r68609;
}
↓
double f(double a, double b, double c) {
        double r68610 = b;
        double r68611 = -r68610;
        double r68612 = r68610 * r68610;
        double r68613 = 3.0;
        double r68614 = a;
        double r68615 = r68613 * r68614;
        double r68616 = c;
        double r68617 = r68615 * r68616;
        double r68618 = r68612 - r68617;
        double r68619 = sqrt(r68618);
        double r68620 = r68611 + r68619;
        double r68621 = r68620 / r68615;
        double r68622 = -3.2444740372448394e-11;
        bool r68623 = r68621 <= r68622;
        double r68624 = -r68618;
        double r68625 = fma(r68610, r68610, r68624);
        double r68626 = r68611 - r68619;
        double r68627 = r68625 / r68626;
        double r68628 = r68627 / r68615;
        double r68629 = -0.5;
        double r68630 = r68616 / r68610;
        double r68631 = r68629 * r68630;
        double r68632 = r68623 ? r68628 : r68631;
        return r68632;
}

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)) < -3.2444740372448394e-11

    1. Initial program 22.6

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

    3. Applied flip-+22.6

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

    4. Simplified21.8

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

    if -3.2444740372448394e-11 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

    1. Initial program 57.1

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \le -3.24447403724483937 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))