Cubic critical, narrow range

Average Error: 28.7 → 16.4

Time: 16.2s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3428323 = b;
        double r3428324 = -r3428323;
        double r3428325 = r3428323 * r3428323;
        double r3428326 = 3.0;
        double r3428327 = a;
        double r3428328 = r3428326 * r3428327;
        double r3428329 = c;
        double r3428330 = r3428328 * r3428329;
        double r3428331 = r3428325 - r3428330;
        double r3428332 = sqrt(r3428331);
        double r3428333 = r3428324 + r3428332;
        double r3428334 = r3428333 / r3428328;
        return r3428334;
}

↓

double f(double a, double b, double c) {
        double r3428335 = b;
        double r3428336 = 174.55343675894656;
        bool r3428337 = r3428335 <= r3428336;
        double r3428338 = r3428335 * r3428335;
        double r3428339 = c;
        double r3428340 = 3.0;
        double r3428341 = a;
        double r3428342 = r3428340 * r3428341;
        double r3428343 = r3428339 * r3428342;
        double r3428344 = r3428338 - r3428343;
        double r3428345 = sqrt(r3428344);
        double r3428346 = r3428344 * r3428345;
        double r3428347 = r3428338 * r3428335;
        double r3428348 = r3428346 - r3428347;
        double r3428349 = r3428335 * r3428345;
        double r3428350 = r3428349 + r3428338;
        double r3428351 = r3428344 + r3428350;
        double r3428352 = r3428348 / r3428351;
        double r3428353 = r3428352 / r3428342;
        double r3428354 = -0.5;
        double r3428355 = r3428339 / r3428335;
        double r3428356 = r3428354 * r3428355;
        double r3428357 = r3428337 ? r3428353 : r3428356;
        return r3428357;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 174.55343675894656

   1. Initial program 15.8

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

   2. Simplified 15.8

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

   4. Applied flip3-- 15.9

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

   5. Simplified 15.2

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

   6. Simplified 15.2

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

   if 174.55343675894656 < b

   1. Initial program 35.3

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

   2. Simplified 35.3

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

   3. Taylor expanded around inf 17.1

      \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 17.1

      \[\leadsto \frac{\frac{-3}{2} \cdot \frac{a \cdot c}{\color{blue}{1 \cdot b}}}{3 \cdot a}\]

   6. Applied times-frac 17.1

      \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(\frac{a}{1} \cdot \frac{c}{b}\right)}}{3 \cdot a}\]

   7. Applied associate-*r* 17.0

      \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot \frac{a}{1}\right) \cdot \frac{c}{b}}}{3 \cdot a}\]

   8. Simplified 17.0

      \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot a\right)} \cdot \frac{c}{b}}{3 \cdot a}\]

   9. Taylor expanded around 0 17.0

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 174.55343675894656:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :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)))