Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, A

Average Error: 3.8 → 0.6

Time: 12.3s

Precision: 64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right) + \left(x \cdot 2 - y \cdot \left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(z \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 2.413987217248999323243296963509637530176 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r492231 = x;
        double r492232 = 2.0;
        double r492233 = r492231 * r492232;
        double r492234 = y;
        double r492235 = 9.0;
        double r492236 = r492234 * r492235;
        double r492237 = z;
        double r492238 = r492236 * r492237;
        double r492239 = t;
        double r492240 = r492238 * r492239;
        double r492241 = r492233 - r492240;
        double r492242 = a;
        double r492243 = 27.0;
        double r492244 = r492242 * r492243;
        double r492245 = b;
        double r492246 = r492244 * r492245;
        double r492247 = r492241 + r492246;
        return r492247;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r492248 = y;
        double r492249 = 9.0;
        double r492250 = r492248 * r492249;
        double r492251 = -3.338893372507657e-32;
        bool r492252 = r492250 <= r492251;
        double r492253 = 27.0;
        double r492254 = a;
        double r492255 = b;
        double r492256 = r492254 * r492255;
        double r492257 = r492253 * r492256;
        double r492258 = x;
        double r492259 = 2.0;
        double r492260 = r492258 * r492259;
        double r492261 = sqrt(r492249);
        double r492262 = z;
        double r492263 = t;
        double r492264 = r492262 * r492263;
        double r492265 = r492261 * r492264;
        double r492266 = r492261 * r492265;
        double r492267 = r492248 * r492266;
        double r492268 = r492260 - r492267;
        double r492269 = r492257 + r492268;
        double r492270 = 2.4139872172489993e-42;
        bool r492271 = r492250 <= r492270;
        double r492272 = r492250 * r492262;
        double r492273 = r492272 * r492263;
        double r492274 = r492260 - r492273;
        double r492275 = r492253 * r492255;
        double r492276 = r492254 * r492275;
        double r492277 = r492274 + r492276;
        double r492278 = sqrt(r492253);
        double r492279 = r492278 * r492256;
        double r492280 = r492278 * r492279;
        double r492281 = r492250 * r492264;
        double r492282 = r492260 - r492281;
        double r492283 = r492280 + r492282;
        double r492284 = r492271 ? r492277 : r492283;
        double r492285 = r492252 ? r492269 : r492284;
        return r492285;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	3.8
Target	2.6
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* y 9.0) < -3.338893372507657e-32

   1. Initial program 7.1

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.8

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]
   4. Using strategy rm

   5. Applied pow1 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot \color{blue}{{b}^{1}}\]

   6. Applied pow1 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot \color{blue}{{27}^{1}}\right) \cdot {b}^{1}\]

   7. Applied pow1 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(\color{blue}{{a}^{1}} \cdot {27}^{1}\right) \cdot {b}^{1}\]

   8. Applied pow-prod-down 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{{\left(a \cdot 27\right)}^{1}} \cdot {b}^{1}\]

   9. Applied pow-prod-down 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{{\left(\left(a \cdot 27\right) \cdot b\right)}^{1}}\]

   10. Simplified 0.8

       \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + {\color{blue}{\left(27 \cdot \left(a \cdot b\right)\right)}}^{1}\]
   11. Using strategy rm

   12. Applied associate-*l* 0.7

       \[\leadsto \left(x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right) + {\left(27 \cdot \left(a \cdot b\right)\right)}^{1}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 0.7

       \[\leadsto \left(x \cdot 2 - y \cdot \left(\color{blue}{\left(\sqrt{9} \cdot \sqrt{9}\right)} \cdot \left(z \cdot t\right)\right)\right) + {\left(27 \cdot \left(a \cdot b\right)\right)}^{1}\]

   15. Applied associate-*l* 0.8

       \[\leadsto \left(x \cdot 2 - y \cdot \color{blue}{\left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(z \cdot t\right)\right)\right)}\right) + {\left(27 \cdot \left(a \cdot b\right)\right)}^{1}\]

   if -3.338893372507657e-32 < (* y 9.0) < 2.4139872172489993e-42

   1. Initial program 0.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.5

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{a \cdot \left(27 \cdot b\right)}\]

   if 2.4139872172489993e-42 < (* y 9.0)

   1. Initial program 7.2

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.7

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]
   4. Using strategy rm

   5. Applied pow1 0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot \color{blue}{{b}^{1}}\]

   6. Applied pow1 0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot \color{blue}{{27}^{1}}\right) \cdot {b}^{1}\]

   7. Applied pow1 0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(\color{blue}{{a}^{1}} \cdot {27}^{1}\right) \cdot {b}^{1}\]

   8. Applied pow-prod-down 0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{{\left(a \cdot 27\right)}^{1}} \cdot {b}^{1}\]

   9. Applied pow-prod-down 0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{{\left(\left(a \cdot 27\right) \cdot b\right)}^{1}}\]

   10. Simplified 0.7

       \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + {\color{blue}{\left(27 \cdot \left(a \cdot b\right)\right)}}^{1}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 0.7

       \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + {\left(\color{blue}{\left(\sqrt{27} \cdot \sqrt{27}\right)} \cdot \left(a \cdot b\right)\right)}^{1}\]

   13. Applied associate-*l* 0.8

       \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + {\color{blue}{\left(\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\right)}}^{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;27 \cdot \left(a \cdot b\right) + \left(x \cdot 2 - y \cdot \left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(z \cdot t\right)\right)\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 2.413987217248999323243296963509637530176 \cdot 10^{-42}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))