quad2p (problem 3.2.1, positive)

Average Error: 1.7 → 0.5

Time: 16.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.0255584716796875:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r1039172 = b_2;
        double r1039173 = -r1039172;
        double r1039174 = r1039172 * r1039172;
        double r1039175 = a;
        double r1039176 = c;
        double r1039177 = r1039175 * r1039176;
        double r1039178 = r1039174 - r1039177;
        double r1039179 = sqrt(r1039178);
        double r1039180 = r1039173 + r1039179;
        double r1039181 = r1039180 / r1039175;
        return r1039181;
}

↓

double f(double a, double b_2, double c) {
        double r1039182 = b_2;
        double r1039183 = -0.0255584716796875;
        bool r1039184 = r1039182 <= r1039183;
        double r1039185 = r1039182 * r1039182;
        double r1039186 = c;
        double r1039187 = a;
        double r1039188 = r1039186 * r1039187;
        double r1039189 = r1039185 - r1039188;
        double r1039190 = sqrt(r1039189);
        double r1039191 = r1039190 - r1039182;
        double r1039192 = r1039191 / r1039187;
        double r1039193 = 0.0;
        double r1039194 = r1039187 * r1039186;
        double r1039195 = r1039193 - r1039194;
        double r1039196 = r1039190 + r1039182;
        double r1039197 = r1039195 / r1039196;
        double r1039198 = r1039197 / r1039187;
        double r1039199 = r1039184 ? r1039192 : r1039198;
        return r1039199;
}

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 < -0.0255584716796875

   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 -0.0255584716796875 < b_2

   1. Initial program 2.9

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

   2. Simplified 2.9

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

   4. Applied p16-flip-- 2.5

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

   6. Applied difference-of-squares 2.9

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

   8. Applied associate-/l* 3.1

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

   10. Applied p16-flip-- 3.2

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

   11. Applied associate-/r/ 3.2

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

   12. Applied associate-/r* 3.2

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

   13. 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

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.0255584716796875:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.0 - a \cdot c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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))