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

Average Error: 3.7 → 1.0

Time: 7.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.2937345159843623 \cdot 10^{-114} \lor \neg \left(t \le 1.66489209295084457 \cdot 10^{-80}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\left(a \cdot 27\right) \cdot b - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r597469 = x;
        double r597470 = 2.0;
        double r597471 = r597469 * r597470;
        double r597472 = y;
        double r597473 = 9.0;
        double r597474 = r597472 * r597473;
        double r597475 = z;
        double r597476 = r597474 * r597475;
        double r597477 = t;
        double r597478 = r597476 * r597477;
        double r597479 = r597471 - r597478;
        double r597480 = a;
        double r597481 = 27.0;
        double r597482 = r597480 * r597481;
        double r597483 = b;
        double r597484 = r597482 * r597483;
        double r597485 = r597479 + r597484;
        return r597485;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r597486 = t;
        double r597487 = -3.293734515984362e-114;
        bool r597488 = r597486 <= r597487;
        double r597489 = 1.6648920929508446e-80;
        bool r597490 = r597486 <= r597489;
        double r597491 = !r597490;
        bool r597492 = r597488 || r597491;
        double r597493 = x;
        double r597494 = 2.0;
        double r597495 = r597493 * r597494;
        double r597496 = y;
        double r597497 = z;
        double r597498 = 9.0;
        double r597499 = r597497 * r597498;
        double r597500 = r597496 * r597499;
        double r597501 = r597500 * r597486;
        double r597502 = r597495 - r597501;
        double r597503 = a;
        double r597504 = 27.0;
        double r597505 = b;
        double r597506 = r597504 * r597505;
        double r597507 = r597503 * r597506;
        double r597508 = r597502 + r597507;
        double r597509 = r597503 * r597504;
        double r597510 = r597509 * r597505;
        double r597511 = r597486 * r597497;
        double r597512 = r597511 * r597496;
        double r597513 = r597498 * r597512;
        double r597514 = r597510 - r597513;
        double r597515 = r597495 + r597514;
        double r597516 = r597492 ? r597508 : r597515;
        return r597516;
}

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	3.7
Target	2.5
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -3.293734515984362e-114 or 1.6648920929508446e-80 < t

   1. Initial program 1.2

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

   3. Applied associate-*l* 1.2

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

   4. Simplified 1.2

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

   6. Applied associate-*l* 1.2

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

   if -3.293734515984362e-114 < t < 1.6648920929508446e-80

   1. Initial program 7.5

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

   3. Applied sub-neg 7.5

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

   4. Applied associate-+l+ 7.5

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

   5. Simplified 7.3

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

   7. Applied associate-*r* 0.6

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.2937345159843623 \cdot 10^{-114} \lor \neg \left(t \le 1.66489209295084457 \cdot 10^{-80}\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(\left(a \cdot 27\right) \cdot b - 9 \cdot \left(\left(t \cdot z\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))