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

Average Error: 3.8 → 2.2

Time: 16.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -5620875659079845081513984:\\ \;\;\;\;\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(27 \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(b \cdot a\right) \cdot 27\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r23626409 = x;
        double r23626410 = 2.0;
        double r23626411 = r23626409 * r23626410;
        double r23626412 = y;
        double r23626413 = 9.0;
        double r23626414 = r23626412 * r23626413;
        double r23626415 = z;
        double r23626416 = r23626414 * r23626415;
        double r23626417 = t;
        double r23626418 = r23626416 * r23626417;
        double r23626419 = r23626411 - r23626418;
        double r23626420 = a;
        double r23626421 = 27.0;
        double r23626422 = r23626420 * r23626421;
        double r23626423 = b;
        double r23626424 = r23626422 * r23626423;
        double r23626425 = r23626419 + r23626424;
        return r23626425;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r23626426 = y;
        double r23626427 = 9.0;
        double r23626428 = r23626426 * r23626427;
        double r23626429 = -5.620875659079845e+24;
        bool r23626430 = r23626428 <= r23626429;
        double r23626431 = a;
        double r23626432 = cbrt(r23626431);
        double r23626433 = r23626432 * r23626432;
        double r23626434 = 27.0;
        double r23626435 = b;
        double r23626436 = r23626434 * r23626435;
        double r23626437 = r23626432 * r23626436;
        double r23626438 = r23626433 * r23626437;
        double r23626439 = x;
        double r23626440 = 2.0;
        double r23626441 = r23626439 * r23626440;
        double r23626442 = t;
        double r23626443 = z;
        double r23626444 = r23626442 * r23626443;
        double r23626445 = r23626428 * r23626444;
        double r23626446 = r23626441 - r23626445;
        double r23626447 = r23626438 + r23626446;
        double r23626448 = r23626428 * r23626443;
        double r23626449 = r23626448 * r23626442;
        double r23626450 = r23626441 - r23626449;
        double r23626451 = r23626435 * r23626431;
        double r23626452 = r23626451 * r23626434;
        double r23626453 = r23626450 + r23626452;
        double r23626454 = r23626430 ? r23626447 : r23626453;
        return r23626454;
}

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	2.8
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if (* y 9.0) < -5.620875659079845e+24

   1. Initial program 8.9

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

   3. Applied associate-*l* 8.8

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

   5. Applied associate-*l* 1.0

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

   7. Applied add-cube-cbrt 1.3

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}\right)} \cdot \left(27 \cdot b\right)\]

   8. Applied associate-*l* 1.3

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(27 \cdot b\right)\right)}\]

   if -5.620875659079845e+24 < (* y 9.0)

   1. Initial program 2.5

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

   2. Taylor expanded around 0 2.4

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -5620875659079845081513984:\\ \;\;\;\;\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\sqrt[3]{a} \cdot \left(27 \cdot b\right)\right) + \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(b \cdot a\right) \cdot 27\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))