Cubic critical, narrow range

Average Error: 28.6 → 16.3

Time: 4.7s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1395.437407789798953672288917005062103271:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r91252 = b;
        double r91253 = -r91252;
        double r91254 = r91252 * r91252;
        double r91255 = 3.0;
        double r91256 = a;
        double r91257 = r91255 * r91256;
        double r91258 = c;
        double r91259 = r91257 * r91258;
        double r91260 = r91254 - r91259;
        double r91261 = sqrt(r91260);
        double r91262 = r91253 + r91261;
        double r91263 = r91262 / r91257;
        return r91263;
}

↓

double f(double a, double b, double c) {
        double r91264 = b;
        double r91265 = 1395.437407789799;
        bool r91266 = r91264 <= r91265;
        double r91267 = r91264 * r91264;
        double r91268 = 3.0;
        double r91269 = a;
        double r91270 = r91268 * r91269;
        double r91271 = c;
        double r91272 = r91270 * r91271;
        double r91273 = r91267 - r91272;
        double r91274 = -r91273;
        double r91275 = fma(r91264, r91264, r91274);
        double r91276 = -r91264;
        double r91277 = sqrt(r91273);
        double r91278 = r91276 - r91277;
        double r91279 = r91275 / r91278;
        double r91280 = r91279 / r91270;
        double r91281 = -0.5;
        double r91282 = r91271 / r91264;
        double r91283 = r91281 * r91282;
        double r91284 = r91266 ? r91280 : r91283;
        return r91284;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 1395.437407789799

   1. Initial program 17.7

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

   3. Applied flip-+ 17.7

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

   4. Simplified 16.8

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

   if 1395.437407789799 < b

   1. Initial program 36.8

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

   2. Taylor expanded around inf 15.9

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1395.437407789798953672288917005062103271:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))