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

Average Error: 3.6 → 0.7

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le -1.509627698880911586338015526628233224088 \cdot 10^{129} \lor \neg \left(\left(y \cdot 9\right) \cdot z \le 2.414314480791499273783363683236492721004 \cdot 10^{188}\right):\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(9 \cdot z\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r675568 = x;
        double r675569 = 2.0;
        double r675570 = r675568 * r675569;
        double r675571 = y;
        double r675572 = 9.0;
        double r675573 = r675571 * r675572;
        double r675574 = z;
        double r675575 = r675573 * r675574;
        double r675576 = t;
        double r675577 = r675575 * r675576;
        double r675578 = r675570 - r675577;
        double r675579 = a;
        double r675580 = 27.0;
        double r675581 = r675579 * r675580;
        double r675582 = b;
        double r675583 = r675581 * r675582;
        double r675584 = r675578 + r675583;
        return r675584;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r675585 = y;
        double r675586 = 9.0;
        double r675587 = r675585 * r675586;
        double r675588 = z;
        double r675589 = r675587 * r675588;
        double r675590 = -1.5096276988809116e+129;
        bool r675591 = r675589 <= r675590;
        double r675592 = 2.4143144807914993e+188;
        bool r675593 = r675589 <= r675592;
        double r675594 = !r675593;
        bool r675595 = r675591 || r675594;
        double r675596 = x;
        double r675597 = 2.0;
        double r675598 = r675596 * r675597;
        double r675599 = r675586 * r675588;
        double r675600 = t;
        double r675601 = r675599 * r675600;
        double r675602 = r675585 * r675601;
        double r675603 = r675598 - r675602;
        double r675604 = 27.0;
        double r675605 = a;
        double r675606 = b;
        double r675607 = r675605 * r675606;
        double r675608 = r675604 * r675607;
        double r675609 = r675603 + r675608;
        double r675610 = r675585 * r675599;
        double r675611 = r675610 * r675600;
        double r675612 = r675598 - r675611;
        double r675613 = r675605 * r675604;
        double r675614 = r675613 * r675606;
        double r675615 = r675612 + r675614;
        double r675616 = r675595 ? r675609 : r675615;
        return r675616;
}

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	2.6
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (* (* y 9.0) z) < -1.5096276988809116e+129 or 2.4143144807914993e+188 < (* (* y 9.0) z)

   1. Initial program 19.0

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

   3. Applied associate-*l* 2.3

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

   5. Applied associate-*l* 1.8

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

   7. Applied associate-*r* 1.9

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

   8. Taylor expanded around 0 1.7

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

   if -1.5096276988809116e+129 < (* (* y 9.0) z) < 2.4143144807914993e+188

   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

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \le -1.509627698880911586338015526628233224088 \cdot 10^{129} \lor \neg \left(\left(y \cdot 9\right) \cdot z \le 2.414314480791499273783363683236492721004 \cdot 10^{188}\right):\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(9 \cdot z\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))