quad2p (problem 3.2.1, positive)

Average Error: 1.7 → 0.5

Time: 17.9s

Precision: 64

Math

\[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.087982177734375:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r1141258 = b_2;
        double r1141259 = -r1141258;
        double r1141260 = r1141258 * r1141258;
        double r1141261 = a;
        double r1141262 = c;
        double r1141263 = r1141261 * r1141262;
        double r1141264 = r1141260 - r1141263;
        double r1141265 = sqrt(r1141264);
        double r1141266 = r1141259 + r1141265;
        double r1141267 = r1141266 / r1141261;
        return r1141267;
}

↓

double f(double a, double b_2, double c) {
        double r1141268 = b_2;
        double r1141269 = -0.087982177734375;
        bool r1141270 = r1141268 <= r1141269;
        double r1141271 = r1141268 * r1141268;
        double r1141272 = c;
        double r1141273 = a;
        double r1141274 = r1141272 * r1141273;
        double r1141275 = r1141271 - r1141274;
        double r1141276 = sqrt(r1141275);
        double r1141277 = r1141276 - r1141268;
        double r1141278 = r1141277 / r1141273;
        double r1141279 = 0.0;
        double r1141280 = r1141273 * r1141272;
        double r1141281 = r1141279 - r1141280;
        double r1141282 = r1141276 + r1141268;
        double r1141283 = r1141281 / r1141282;
        double r1141284 = r1141283 / r1141273;
        double r1141285 = r1141270 ? r1141278 : r1141284;
        return r1141285;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b_2 < -0.087982177734375

   1. Initial program 0.4

      \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

   2. Simplified 0.4

      \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]

   if -0.087982177734375 < b_2

   1. Initial program 2.6

      \[\frac{\left(\frac{\left(-b_2\right)}{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)}\right)}{a}\]

   2. Simplified 2.6

      \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}{a}}\]
   3. Using strategy rm

   4. Applied p16-flip-- 2.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)\right) - \left(b_2 \cdot b_2\right)\right)}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}}{a}\]
   5. Using strategy rm

   6. Applied difference-of-squares 2.7

      \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right) \cdot \left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)\right)}}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}{a}\]

   7. Applied associate-/l* 2.8

      \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) - b_2\right)}\right)}\right)}}{a}\]
   8. Using strategy rm

   9. Applied p16-flip-- 3.0

      \[\leadsto \frac{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)\right) - \left(b_2 \cdot b_2\right)\right)}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}}\right)}\right)}{a}\]

   10. Applied associate-/r/ 3.0

       \[\leadsto \frac{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)\right) - \left(b_2 \cdot b_2\right)\right)}\right) \cdot \left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)\right)}}\right)}{a}\]

   11. Applied associate-/r* 3.0

       \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\frac{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}{\left(\left(\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right) \cdot \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)\right) - \left(b_2 \cdot b_2\right)\right)}\right)}\right)}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}}{a}\]

   12. Simplified 0.6

       \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(0.0\right) - \left(a \cdot c\right)\right)}}{\left(\frac{\left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(c \cdot a\right)\right)}\right)}{b_2}\right)}\right)}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.087982177734375:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/.p16 (+.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))