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

Average Error: 3.6 → 1.1

Time: 16.2s

Precision: 64

Math

\[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9.0 \le -2.571496331294558 \cdot 10^{+88}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(t \cdot z\right)\right) + a \cdot \left(b \cdot 27.0\right)\\ \mathbf{elif}\;y \cdot 9.0 \le 3.712720064367962 \cdot 10^{-126}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(\left(y \cdot z\right) \cdot t\right)\right) + b \cdot \left(27.0 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(t \cdot z\right)\right) + a \cdot \left(b \cdot 27.0\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r41025416 = x;
        double r41025417 = 2.0;
        double r41025418 = r41025416 * r41025417;
        double r41025419 = y;
        double r41025420 = 9.0;
        double r41025421 = r41025419 * r41025420;
        double r41025422 = z;
        double r41025423 = r41025421 * r41025422;
        double r41025424 = t;
        double r41025425 = r41025423 * r41025424;
        double r41025426 = r41025418 - r41025425;
        double r41025427 = a;
        double r41025428 = 27.0;
        double r41025429 = r41025427 * r41025428;
        double r41025430 = b;
        double r41025431 = r41025429 * r41025430;
        double r41025432 = r41025426 + r41025431;
        return r41025432;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r41025433 = y;
        double r41025434 = 9.0;
        double r41025435 = r41025433 * r41025434;
        double r41025436 = -2.571496331294558e+88;
        bool r41025437 = r41025435 <= r41025436;
        double r41025438 = x;
        double r41025439 = 2.0;
        double r41025440 = r41025438 * r41025439;
        double r41025441 = t;
        double r41025442 = z;
        double r41025443 = r41025441 * r41025442;
        double r41025444 = r41025435 * r41025443;
        double r41025445 = r41025440 - r41025444;
        double r41025446 = a;
        double r41025447 = b;
        double r41025448 = 27.0;
        double r41025449 = r41025447 * r41025448;
        double r41025450 = r41025446 * r41025449;
        double r41025451 = r41025445 + r41025450;
        double r41025452 = 3.712720064367962e-126;
        bool r41025453 = r41025435 <= r41025452;
        double r41025454 = r41025433 * r41025442;
        double r41025455 = r41025454 * r41025441;
        double r41025456 = r41025434 * r41025455;
        double r41025457 = r41025440 - r41025456;
        double r41025458 = r41025448 * r41025446;
        double r41025459 = r41025447 * r41025458;
        double r41025460 = r41025457 + r41025459;
        double r41025461 = r41025453 ? r41025460 : r41025451;
        double r41025462 = r41025437 ? r41025451 : r41025461;
        return r41025462;
}

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	3.6
Target	2.4
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + a \cdot \left(27.0 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* y 9.0) < -2.571496331294558e+88 or 3.712720064367962e-126 < (* y 9.0)

   1. Initial program 6.7

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 1.3

      \[\leadsto \left(x \cdot 2.0 - \color{blue}{\left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27.0\right) \cdot b\]
   4. Using strategy rm

   5. Applied associate-*l* 1.3

      \[\leadsto \left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)\right) + \color{blue}{a \cdot \left(27.0 \cdot b\right)}\]

   if -2.571496331294558e+88 < (* y 9.0) < 3.712720064367962e-126

   1. Initial program 0.9

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

   2. Taylor expanded around inf 0.9

      \[\leadsto \left(x \cdot 2.0 - \color{blue}{9.0 \cdot \left(t \cdot \left(z \cdot y\right)\right)}\right) + \left(a \cdot 27.0\right) \cdot b\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9.0 \le -2.571496331294558 \cdot 10^{+88}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(t \cdot z\right)\right) + a \cdot \left(b \cdot 27.0\right)\\ \mathbf{elif}\;y \cdot 9.0 \le 3.712720064367962 \cdot 10^{-126}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(\left(y \cdot z\right) \cdot t\right)\right) + b \cdot \left(27.0 \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(t \cdot z\right)\right) + a \cdot \left(b \cdot 27.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))