Cubic critical, narrow range

Average Error: 28.9 → 16.4

Time: 57.2s

Precision: 64

• unused variable d

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 157.03145246793306:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r16546245 = b;
        double r16546246 = -r16546245;
        double r16546247 = r16546245 * r16546245;
        double r16546248 = 3.0;
        double r16546249 = a;
        double r16546250 = r16546248 * r16546249;
        double r16546251 = c;
        double r16546252 = r16546250 * r16546251;
        double r16546253 = r16546247 - r16546252;
        double r16546254 = sqrt(r16546253);
        double r16546255 = r16546246 + r16546254;
        double r16546256 = r16546255 / r16546250;
        return r16546256;
}

↓

double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r16546257 = b;
        double r16546258 = 157.03145246793306;
        bool r16546259 = r16546257 <= r16546258;
        double r16546260 = r16546257 * r16546257;
        double r16546261 = c;
        double r16546262 = 3.0;
        double r16546263 = a;
        double r16546264 = r16546262 * r16546263;
        double r16546265 = r16546261 * r16546264;
        double r16546266 = r16546260 - r16546265;
        double r16546267 = sqrt(r16546266);
        double r16546268 = r16546266 * r16546267;
        double r16546269 = r16546260 * r16546257;
        double r16546270 = r16546268 - r16546269;
        double r16546271 = r16546257 * r16546267;
        double r16546272 = r16546271 + r16546260;
        double r16546273 = r16546267 * r16546267;
        double r16546274 = r16546272 + r16546273;
        double r16546275 = r16546264 * r16546274;
        double r16546276 = r16546270 / r16546275;
        double r16546277 = -0.5;
        double r16546278 = r16546261 / r16546257;
        double r16546279 = r16546277 * r16546278;
        double r16546280 = r16546259 ? r16546276 : r16546279;
        return r16546280;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

1. Split input into 2 regimes

2. if b < 157.03145246793306

   1. Initial program 15.7

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

   2. Simplified 15.7

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

   4. Applied flip3-- 15.8

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

   5. Applied associate-/l/ 15.8

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

   6. Simplified 15.1

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

   if 157.03145246793306 < 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

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 157.03145246793306:\\ \;\;\;\;\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(3 \cdot a\right) \cdot \left(\left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (a b c d)
  :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)))