Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Average Error: 1.9 → 1.2

Time: 8.1s

Precision: 64

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

Math

\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.67032308676182466 \cdot 10^{79}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r701481 = x;
        double r701482 = y;
        double r701483 = z;
        double r701484 = r701482 * r701483;
        double r701485 = r701481 + r701484;
        double r701486 = t;
        double r701487 = a;
        double r701488 = r701486 * r701487;
        double r701489 = r701485 + r701488;
        double r701490 = r701487 * r701483;
        double r701491 = b;
        double r701492 = r701490 * r701491;
        double r701493 = r701489 + r701492;
        return r701493;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r701494 = a;
        double r701495 = -1.6703230867618247e+79;
        bool r701496 = r701494 <= r701495;
        double r701497 = z;
        double r701498 = b;
        double r701499 = t;
        double r701500 = fma(r701497, r701498, r701499);
        double r701501 = y;
        double r701502 = x;
        double r701503 = fma(r701501, r701497, r701502);
        double r701504 = fma(r701494, r701500, r701503);
        double r701505 = r701501 * r701497;
        double r701506 = r701502 + r701505;
        double r701507 = r701499 * r701494;
        double r701508 = r701506 + r701507;
        double r701509 = r701494 * r701497;
        double r701510 = r701509 * r701498;
        double r701511 = r701508 + r701510;
        double r701512 = r701496 ? r701504 : r701511;
        return r701512;
}

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	1.9
Target	0.3
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -1.6703230867618247e+79

   1. Initial program 6.2

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), \mathsf{fma}\left(y, z, x\right)\right)}\]

   if -1.6703230867618247e+79 < a

   1. Initial program 1.3

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.67032308676182466 \cdot 10^{79}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), \mathsf{fma}\left(y, z, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))