Cubic critical, medium range

Average Error: 44.0 → 10.0

Time: 18.7s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -0.004961147766964667943367928160114388447255:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b \cdot \left(b \cdot b\right)}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3}, \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} + b, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2858505 = b;
        double r2858506 = -r2858505;
        double r2858507 = r2858505 * r2858505;
        double r2858508 = 3.0;
        double r2858509 = a;
        double r2858510 = r2858508 * r2858509;
        double r2858511 = c;
        double r2858512 = r2858510 * r2858511;
        double r2858513 = r2858507 - r2858512;
        double r2858514 = sqrt(r2858513);
        double r2858515 = r2858506 + r2858514;
        double r2858516 = r2858515 / r2858510;
        return r2858516;
}

↓

double f(double a, double b, double c) {
        double r2858517 = b;
        double r2858518 = r2858517 * r2858517;
        double r2858519 = 3.0;
        double r2858520 = a;
        double r2858521 = r2858519 * r2858520;
        double r2858522 = c;
        double r2858523 = r2858521 * r2858522;
        double r2858524 = r2858518 - r2858523;
        double r2858525 = sqrt(r2858524);
        double r2858526 = -r2858517;
        double r2858527 = r2858525 + r2858526;
        double r2858528 = r2858527 / r2858521;
        double r2858529 = -0.004961147766964668;
        bool r2858530 = r2858528 <= r2858529;
        double r2858531 = r2858522 * r2858520;
        double r2858532 = r2858531 * r2858519;
        double r2858533 = r2858518 - r2858532;
        double r2858534 = sqrt(r2858533);
        double r2858535 = r2858533 * r2858534;
        double r2858536 = r2858517 * r2858518;
        double r2858537 = r2858535 - r2858536;
        double r2858538 = r2858534 + r2858517;
        double r2858539 = fma(r2858534, r2858538, r2858518);
        double r2858540 = r2858537 / r2858539;
        double r2858541 = r2858540 / r2858521;
        double r2858542 = -0.5;
        double r2858543 = r2858522 / r2858517;
        double r2858544 = r2858542 * r2858543;
        double r2858545 = r2858530 ? r2858541 : r2858544;
        return r2858545;
}

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)) < -0.004961147766964668

   1. Initial program 20.0

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

   3. Applied flip3-+ 20.1

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

   4. Simplified 19.4

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

   5. Simplified 19.4

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

   if -0.004961147766964668 < (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))

   1. Initial program 49.9

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

   2. Taylor expanded around inf 7.6

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \le -0.004961147766964667943367928160114388447255:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(c \cdot a\right) \cdot 3\right) \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} - b \cdot \left(b \cdot b\right)}{\mathsf{fma}\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3}, \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 3} + b, b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))