Quadratic roots, medium range

Average Error: 43.8 → 10.4

Time: 18.6s

Precision: 64

\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -1.7506913033883893 \cdot 10^{-08}:\\ \;\;\;\;\frac{b \cdot b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1640305 = b;
        double r1640306 = -r1640305;
        double r1640307 = r1640305 * r1640305;
        double r1640308 = 4.0;
        double r1640309 = a;
        double r1640310 = r1640308 * r1640309;
        double r1640311 = c;
        double r1640312 = r1640310 * r1640311;
        double r1640313 = r1640307 - r1640312;
        double r1640314 = sqrt(r1640313);
        double r1640315 = r1640306 + r1640314;
        double r1640316 = 2.0;
        double r1640317 = r1640316 * r1640309;
        double r1640318 = r1640315 / r1640317;
        return r1640318;
}

↓

double f(double a, double b, double c) {
        double r1640319 = b;
        double r1640320 = r1640319 * r1640319;
        double r1640321 = 4.0;
        double r1640322 = a;
        double r1640323 = r1640321 * r1640322;
        double r1640324 = c;
        double r1640325 = r1640323 * r1640324;
        double r1640326 = r1640320 - r1640325;
        double r1640327 = sqrt(r1640326);
        double r1640328 = -r1640319;
        double r1640329 = r1640327 + r1640328;
        double r1640330 = 2.0;
        double r1640331 = r1640330 * r1640322;
        double r1640332 = r1640329 / r1640331;
        double r1640333 = -1.7506913033883893e-08;
        bool r1640334 = r1640332 <= r1640333;
        double r1640335 = r1640327 * r1640327;
        double r1640336 = r1640320 - r1640335;
        double r1640337 = r1640328 - r1640327;
        double r1640338 = r1640331 * r1640337;
        double r1640339 = r1640336 / r1640338;
        double r1640340 = -r1640324;
        double r1640341 = r1640340 / r1640319;
        double r1640342 = r1640334 ? r1640339 : r1640341;
        return r1640342;
}

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) (* (* 4 a) c)))) (* 2 a)) < -1.7506913033883893e-08

   1. Initial program 22.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied flip-+ 22.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   4. Applied associate-/l/ 22.2

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]

   if -1.7506913033883893e-08 < (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))

   1. Initial program 54.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Taylor expanded around inf 4.5

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

   3. Simplified 4.5

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a} \le -1.7506913033883893 \cdot 10^{-08}:\\ \;\;\;\;\frac{b \cdot b - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))